# The RIT Discrete and Computational Mathematics Seminar

## Overview

Welcome to the Discrete and Computational Mathematics Seminar at RIT (DisCoMathS)! We are a big tent seminar series for everything discrete: graph theory, combinatorics, combinatorial optimization, applications of discrete mathematics, and computational aspects of all these subjects. During the Spring 2023 semester the seminar meets most weeks (see the schedule below for the exact dates) in Thomas Gosnell Hall (GOS-2154). All talks are scheduled from 3:30PM until 4:30PM Rochester time on Tuesdays (unless otherwise noted). For seminar announcements, and schedule updates, be sure to subscribe to the mailing list (see below).

DisCoMathS is open to the public, and everyone is welcome to attend. Graduate and undergraduate students are especially welcome.

The seminar is organized this semester by Brendan Rooney. If you would like to give a seminar, please contact discomaths@rit.edu.

## Mailing List

Seminar communications (announcements, scheduling polls, etc.) will be sent via the DisCoMathS mailing list. Join now!

## Past Seminars

Tuesday, February 14

* Speaker: *Brendan Rooney

*Cyclotomic Matrices and Graphs with $q=2$*

**Title:*** Abstract: *We survey a sequence of papers by Smyth, McKee, Greaves, and Taylor (see [1], [2], [3], [4], [5], [6]) on Lehmer's Conjecture and cyclotomic matrices. Our focus is the graphs corresponding to maximal indecomposable cyclotomic matrices (GMICs for short).

For a graph $G$, the value $q(G)$ is the smallest number of distinct eigenvalues of a symmetric matrix $M$ whose off-diagonal zero pattern matches that of the adjacency matrix of $G$ exactly. Recent work on sparse graphs, and $4$-regular graphs, with $q(G)=2$ reveals that most of these $q=2$ graphs correspond to GMICs. This talk is part of an attempt to connect these bodies of work, and will contain open questions.

Tuesday, February 28

* Speaker: *Zohair Hassan (RIT)

*On the hardness of $(P_3, H)$-Non-Arrowing*

**Title:*** Abstract: *Graph arrowing is concerned with determining which monochromatic subgraphs are unavoidable when coloring a given graph. The complexity of various arrowing problems has been studied for decades, particularly when the subgraphs to be avoided are fixed. Some cases have been shown to be solvable in polynomial time or NP-complete. We focus on the latter case.

In this presentation, we will focus on $(P_3, H)$-Non-Arrowing; determining whether it is possible to color a given graph's edges red or blue such that there are no red $P_3$'s or blue $H$'s, where $P_3$ is the path graph on three vertices, and $H$ is a fixed graph. In particular, we will observe how structural properties (e.g., $k$-connectivity) of $H$ allow us to construct gadgets to reduce a variant of SAT to the $(P_3, H)$-Non-Arrowing problem. This provides hardness results for an infinitely large family of $H$.

Tuesday, March 21

* Speaker: *Matt Coppenbarger

*Oodles and oodles of googols: Iterations of the Words-to-Numbers Function*

**Title:*** Abstract:* The Words-to-Numbers function takes a nonnegative integer and maps it to the number of characters required to spell the number in English. For each $k=0,1,\ldots,8$, we determine the minimal nonnegative integer $n$ such that the iterates of $n$ converge to the fixed point (through the Words-to-Numbers function) in exactly $k$ iterations. Our travels take us from some simple answers, to an etymology of the names of large numbers, to Conway/Guy's established system representing the name of any integer, and finally use generating functions to arrive at a very large number.

Tuesday, March 28

* Speaker: *Kristijan Tabak

*Cubes of designs*

**Title:*** Abstract:* A symmetric design is a combinatorial object represented as a matrix with entries 0 and 1, where any two rows and any two columns have equal number of ones on respective locations. Historically, symmetric designs were studied in genetics and in designs of pharmaceutical experiments. Afterwards, it became a mainstream part of mathematics.

In this talk we shall represent new developments in a theory of multidimensional symmetric designs that was originated by dr. Krčadinac, dr. Pavčević and dr. Tabak. In brief, we study 3-dimensional, 4-dimensional etc. cubes of (0,1) matrices that preserve a great deal of symmetry inherited from a basic 2-dimensional case.

Applications of cubes of designs are possible in a complex setting of statistical tests, neural networks, optimization problems etc.

Tuesday, April 4

* Speaker: *Kristijan Tabak

*Surprising dualities of incidence structures across finite vector spaces*

**Title:*** Abstract:* A natural generalization of a symmetric design is usually made within the frame of a vector space over the finite field. With this in mind, we present new dualities in a ‘min-max’ way, meaning that some minimal substructures in a vector space (over finite field) are deeply related with the appropriate maximal structures in the same ambient space. Both ‘extremes’ are constructed using generalizations of a symmetric design and also, what it seems now, not related with classical concepts of residual or derived designs. This is still work in progress. We believe that this could yield to some new algebraic identities that may be interpreted as ‘Mobius like’ inversion formulas that we have seen in number theory.

Tuesday, April 11

* Speaker: *Darren Narayan

*Failed Zero Forcing Numbers of Graphs*

**Title:*** Abstract:* Given a graph $G$, the zero forcing number of $G$, $Z(G)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the forcing rule results in all vertices being in $S$. The forcing rule is: if a vertex $v$ is in $S$, and exactly one neighbor $u$ of $v$ is not in $S$, then $u$ is added to $S$ in the next iteration. Zero forcing numbers have attracted great interest over the past 15 years and have been well studied. In this paper, we investigate the largest size of a set $S$ that does not force all of the vertices in a graph to be in $S$. This quantity is known as the failed zero forcing number of a graph and will be denoted by $F(G)$. We present new results involving this parameter.

Tuesday, April 18

* Speaker: *Shahla Nasserasr

*On the Inverse Eigenvalue Problem and Distinctness of Eigenvalues*

**Title:*** Abstract:* Consider the set all $n\times n$ real symmetric matrices where all of them follow a fixed pattern of zero-nonzero entries on their off-diagonal entries. The inverse eigenvalue problem then asks to determine all possible spectra of such matrices. The question is difficult and one way to relax it is to study the minimum number of distinct eigenvalues instead of the values of the eigenvalues. This parameter is denoted by $q(G)={\rm min}\{q(A)|A \in \mathcal{S}(G)\}$ where $G$ is a graph that represents the pattern of the set of matrices. In this presentation, I will review some of the results and the techniques from recent developments about $q(G)$.

Wednesday, September 14

* Speaker: *Brendan Rooney

*How to Draw a Graph Pt 1*

**Title:*** Abstract:* We review some properties of the Laplacian matrix of a graph, and show how it can be used to draw graphs in the plane.

Wednesday, September 21

* Speaker: *Brendan Rooney

*How to Draw a Graph Pt 2*

**Title:*** Abstract:* We continue our discussion of graph drawings using the Laplacian matrix. In this seminar we will see Tutte's method for drawing 3-connected planar graphs, and Steinitz's Theorem on convex polyhedra in $\mathbb{R}^3$.

Wednesday, September 28

* Speaker: *Matthew Coppenbarger

*The Principle of Parity in Analyzing Puzzles*

**Title:*** Abstract: *The mathematical principle of parity is the simple and obvious fact that an even number is not equal to an odd number. It is a powerful tool that is usually restricted for use in impossibility proofs, but it can also be used as a way to make some very challenging puzzles where the solutions utilize the principle of parity.

This talk will cover many puzzles, starting with the well-known *The Puzzle of 15*, the impossibility of the *14-15 Puzzle*, and a few other sliding block puzzles. In addition, half a dozen construction puzzles using various polyform pieces will be introduced, including a more elaborate version of the *Soma Cube*. Lastly, a brief discussion of the *Eternity Puzzle*. This last puzzle originated in the Britain and offered a $1.4 million prize to the first person or persons to produce a complete solution.

Solutions of most puzzles will not be provided, but clues on how to find the solution will be given. A number of the discussed puzzles will be available after the talk. No special knowledge of mathematics is needed to understand the material presented.

Wednesday, October 12

* Speaker: *Shahla Nasserasr

*Zero Forcing Game and Some of Its Variants*

**Title:*** Abstract: *Zero forcing of a graph is a color-change rule on the vertices of the graph using two colors. This a one-player game and the goal is to color all of the vertices with the same color (following the rules). The color-change rule and the number of players can vary depending the underlying problem and matrices. In this talk we will review some variants of zero forcing and their properties.

Wednesday, November 2

* Speaker: *Bonnie Jacob

*Two problems in power domination*

**Title:****Abstract:** What is the most efficient way to successfully place phasor measurement units (PMUs) on an electric power grid? The graph theoretic approach to modeling this problem is known as power domination. Mathematically speaking, power domination has a great deal in common with the well-known graph theory areas of domination and zero forcing, and interestingly, was introduced earlier than zero forcing. In this talk, I will talk about failed power domination which is the study of how many PMUs can be placed on the grid without effectively monitoring the grid, as well as power domination reconfiguration, which is the investigation of when one PMU configuration can be changed into another under certain constraints.

Wednesday, November 16

* Speaker: *Hossein Shahmohamad

*Graphs and Their Potent Energy Drinks*

**Title:****Abstract: **Energy of a graph was defined by Gutman in 1978 and originates from theoretical chemistry. For an $n$-vertex graph $G$ with adjacency matrix $A$ having eigenvalues $\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_n$, the energy $E(G)$ is defined as the sum of absolute values of all eigenvalues, i.e., $E(G)=\sum_{i=1}^n|\lambda_i|$. This is related to the total $\pi$-electron energy in a molecule represented by a graph. Origins of this topic are found in Hückel molecular orbital theory (HMO), $\pi$-electrons and their total energy $E_{\pi}$, molecular orbital energy levels $E_j$, the HMO Hamiltonian operator. The term “energy” comes from quantum Chemistry. The study of graph energy is deeply related to modeling of spread of epidemics, properties of proteins, the search for the genetic causes of Alzheimer disease, and entropy. In this talk, we compile and summarize many of the recent discoveries and progress made on this topic.

February 9

* Speaker: *Brendan Rooney

*Graphs with Few Eigenvalues*

**Title:*** Abstract:*A graph $G$ has several matrices associated with it: its adjacency matrix; its Laplacian matrix; its Laplacian matrix; and, its Seidel matrix. All of these are well-studied. We will look at graphs for which these matrices have a small number of distinct eigenvalues. We will see some of the classical theory of strongly regular graphs. The results of Haemers and Omidi on eigenvalues of the universal adjacency matrix of $G$ will be discussed.

February 16

* Speaker: *Hannah Miller

*Enumerating Nontrivial Knot Mosaics with SAT*

**Title:*** Abstract:*Mathematical knots are interesting topological objects. Using simple arcs, lines, and crossings drawn on eleven possible tiles, knot mosaics are a representation of knots on a mosaic board. Our contribution is using SAT solvers as a tool for enumerating nontrivial knot mosaics. By encoding constraints for local knot mosaic properties, we computationally reduce the search space for small mosaics by factors of up to 6600. Our future research directions include attacking larger mosaics with parallel SAT techniques and studying the behavior of ALLSAT solvers on our knot mosaics SAT formula.

March 2

* Speaker: *Brendan Rooney

*Equiangular Lines*

**Title:*** Abstract:*We survey some of the theory on equiangular sets of lines. Our focus will be on Seidel matrices, with an eye to finding useful expressions of matrices in $\mathcal{S}(K_n)$ as Gram matrices. We will also see connections to strongly regular graphs.

March 16

* Speaker: *Zohair Hassan

*On the Hardness of Graph Arrowing: A Combinatorial Computing Approach Part 1*

**Title:*** Abstract:* Graph arrowing is concerned with determining which monochromatic subgraphs are unavoidable when coloring a given graph. The complexity of various arrowing problems has been studied for decades, particularly when the subgraphs to be avoided are fixed. Some cases have been shown to be solvable in polynomial time or to be coNP-complete. We focus on the latter case. Previous hardness proofs typically rely on ad-hoc, laborious constructions of "gadgets." We propose searching for gadgets computationally. These gadgets will allow us to simulate variants of SAT, thus showing coNP-hardness.

In this presentation, we will focus on $P_3P_5$-Arrowing; determining whether it is possible to color a given graph's edges red or blue such that there are no red $P_3$'s or blue $P_5$'s, where $P_n$ is the path graph on $n$ vertices. We will show that $P_3P_5$-Arrowing is coNP-complete by discussing the properties of gadgets that may be used in a proof.

March 23

* Speaker: *Zohair Hassan

*On the Hardness of Graph Arrowing: A Combinatorial Computing Approach Part 2*

**Title:*** Abstract: *We previously explored $P_3P_5$-Arrowing and how one could prove its coNP-hardness with the right gadgets. In this presentation, we will explore how to find these gadgets computationally. Our approach is based on enumerating over a set of generated candidate graphs until we find one that meets the requirements of our gadget.

We will discuss a general framework that can be implemented to generate candidate graphs, and how this framework can be made more efficient. This will be followed by a presentation of the results, showcasing the gadgets found and the complete proof. Finally, we will discuss the applicability of this approach to other graph arrowing problems.

March 30

* Speaker: *John Mooney

*Numerically Balanced Dice*

**Title:*** Abstract: *Dice can only be fair in theory due to various physical imperfections that inevitably occur during manufacturing processes. A die is considered

*numerically balanced*despite physical imperfections when the numbers assigned to its faces satisfy sets of linear constraints to preserve a fair die's expected value. The existence of numerically balanced dice was first confirmed in Bosch et al. using integer programming techniques to find balanced numberings of the d20, d30, and d120. We explore how to answer questions regarding uniqueness (up to isomorphism) of some isohedra suitable for use as fair dice. In this talk, we discuss the combinatorics involved to make the problem more computationally tractable and as well as approaches that eliminate the need for computation altogether. We present 5 new non-isomorphic numerically balanced numberings of the d20 (regular icosahedron).

April 13

* Speaker: *Matt Coppenbarger

*Maximal Heights of the Bulgarian Solitaire Function*

**Title:*** Abstract:* The Bulgarian Solitaire function, introduced nearly 40 years ago by Martin Gardner, is a function on integer partitions. It is best modeled by a deck of cards (or any collection of indistinguishable counters - coins work too). The cards are divided into any number of stacks. The function takes a single counter from each stack and forms a new stack (and so stacks with a single counter disappear when the counter is removed). The process is iterated over and over to see what happens. The talk will go through the main results and the open question that I've been working on for a few years.

September 17

* Speaker: *Shahla Nasserasr

*Rigid Linkages and Eigenvalues*

**Title:*** Abstract: *Given an assignment of colours blue and white to the vertices of graph $G$, the zero forcing rule determines a subset of white vertices to be re-coloured blue. The zero forcing process applies this rule in discrete time-steps until no more changes are possible. Rigid linkage forcing is defined similarly to the standard zero forcing process. Rigid linkages are used to provide more information on the multiplicities of the eigenvalues, and the relationship between the eigenvalues of matrices related to graphs.

In this talk, some properties and applications of the rigid linkage forcing process will be discussed.

September 24

* Speaker: *Brendan Rooney

*Combinatorial Orthogonality and Graphs with $q=2$*

**Title:*** Abstract:* Two vectors are

*combinatorially orthogonal*if the size of the intersection of their supports is not 1. An $n\times n$ symmetric matrix is

*combinatorially orthogonal*if every pair of rows, or columns, is combinatorially orthogonal. Every orthogonal pair of vectors is combinatorially orthogonal, and every orthogonal matrix is combinatorially orthogonal. For a graph $G$, we give a simple structural condition that guarantees the existence of a combinatorially orthogonal matrix in $\mathcal{S}(G)$ (the set of symmetric matrices with the same off-diagonal zero-pattern as the adjacency matrix of $G$). We will see how this connects with work done by Reid and Thomassen on graphs for which every path of length $r$ is contained in a cycle of length $s$. Finally we discuss implications for finding $q(G)$, and characterizing graphs with $q=2$.

October 1

* Speaker: *Bonnie Jacob

*>Bringing Together Zero Forcing and Failed Zero Forcing: The Zero Forcing Span of a Graph*

**Title:*** Abstract:*Between the zero forcing number of a graph $\mbox{Z}(G)$ and the failed zero forcing number $\mbox{F}(G)$ is where anything can happen. More specifically, if $\mbox{Z}(G) \leq k < \mbox{F}(G)$, there exist sets of cardinality $k$ that are zero forcing sets, and sets of cardinality $k$ that are not. For some graphs, there are many such cardinalities (or equivalently, the difference between $\mbox{Z}(G)$ and $\mbox{F}(G)$ is large), while for others, there are none. We call the number of such cardinalities the

*zero forcing span*of a graph. In this talk, we connect results on $\mbox{Z}(G)$ with results on $\mbox{F}(G)$ to describe graphs that have various zero forcing spans, providing characterizations of some extreme values.

October 8

* Speaker:* Kristijan Tabak (RIT Croatia)

*Binary Fano Plane and Automorphisms of Order 7*

**Title:*** Abstract: *The binary Fano plane is a combinatorial design embedded in 7-dimensional finite vector space over $GF(2)$. The blocks are made of 3-dimensional subspaces so that any 2-dimensional space is a subspace of just one block. It is still unknown whether such a design exists. What is known is that (among others) an automorphism of order 7 can’t operate on a set of blocks. This was proved by extensive use of computer power.

Here we present the first algebraic computer-free proof of that fact. The methods used in the proof involve group theory, character theory and calculations in various group rings.

October 15

* Speaker:* Peter Maceli (Ithaca College)

*Chi-boundedness*

**Title:*** Abstract: *If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs? In the early 1980's András Gyárfás made a number of challenging conjectures about this. In this talk, we will give a brief survey of how these questions seek to generalize the class of perfect graphs, along with some recent results.

October 22

* Speaker:* Kristijan Tabak (RIT Croatia)

*On $p$-Groups with Minimal Subgroup Centralizers - New Results*

**Title:*** Abstract: *We present new results on a problem of $CZ_p$-groups (i.e., groups $G$ for which the centralizer of any subgroup is the center of $G$). Such groups are highly nonabelian. The results presented here will be a natural continuation of previously published results where we have proved that such groups have at least $p^5$ elements.

**We make new connections between $CZ_p$-groups and groups of maximal class.**

October 29

* Speaker:* Adam Giammarese (RIT)

*Decision Tree-Based Parameter-Free Method for Chaotic Time Series Forecasting*

**Title:*** Abstract: *Forecasting the future evolution of chaotic data-based systems is a crucial, but challenging practical problem. Existing solutions - such as Recurrent Neural Networks (RNNs) or Reservoir Computing (RC) - have been shown to be effective methods of forecasting time series, but require a slew of parameters and hyperparameters. In this work, we discuss a mostly parameter-free machine learning approach to chaotic time series forecasting and feature selection in the form of an Extra-Trees Regressor (ETR), which utilizes an ensemble of regression trees. We develop a method involving ETR that provides a notable performance in forecasting the future evolution of chaotic time series with limited information about the system itself. Using temporal systems such as the Rossler System, we demonstrate the efficacy of the developed forecasting method. The ETR-based forecast method is also applied to various (and more difficult to predict) systems, such as the spatio-temporal Kuramoto–Sivashinsky system. In comparison to the existing RNN and RC methods, we observe that our method provides comparable (if not impressive) performance while requiring far fewer parameters and hyperparameters that traditionally make the former systems difficult to use in practice.

November 5

* Speaker: *Quinn Kolt

*Constructing Unitary Fusion Category Representations with the Jellyfish Algorithm*

**Title:*** Abstract:* The graphical calculus is a modern technique for classifying and constructing “unitary fusion” categories, categories which model elementary particles that merge and split. In particular, if one can obtain enough diagrammatic relations to evaluate all “closed” diagrams, this graphical calculus uniquely describes the category. However, in order to prove existence, one must also find a representation of the category. Cuntz algebras, a type of C*-algebra, are a natural home for the representations of many fusion categories. The Jellyfish algorithm allows us to use diagrams to find what these Cuntz algebra representations should be in an intuitive way. Using this computation, we can then prove existence by showing that this construction is indeed a well-defined representation in the associated Cuntz algebra. In this talk, I will apply this process on the unitary fusion category with Fibonacci fusion rules.

November 12

* Speaker: *Yutong Wu

*Failed Positive Semidefinite Zero Forcing*

**Title:*** Abstract:* Given a simple, undirected graph $G$, consider each vertex in $V(G)$ as either “filled” or “unfilled”. Let $S$ be the set of vertices that are filled. The positive semidefinite zero forcing rule is as follows:

- Consider each component of $G-S$.
- For each component $G_i$ of $G-S$, consider $G_i+A$, where $A$ is the set neighbors of the vertices in $G_i$ from $S$.
- Apply zero forcing color change rule to $G_i+A$. That is, an unfilled vertex $v$ is forced to be filled if it is the only unfilled neighbor of a filled vertex.

The maximum size of a set of filled vertices that fails to fill all vertices of $G$ while applying the positive semidefinite zero forcing rule, denoted by $F^+(G)$, is called the failed positive semidefinite zero forcing number. We will discuss the parameter $F^+(G)$ for different types of graphs, as well as characterization of graphs with large $F^+(G)$ and graphs with small $F^+(G)$.

November 19

* Speaker: *Matt Coppenbarger

*An Impartial Combinatorial Game on a $3\times 3$ Board with Magic Square Constraints*

**Title:*** Abstract:*Two players, Alice and Bob (with Alice moving first), alternate on choosing an integer from $0$ to $n$ (inclusive) and entering the number into an empty space on an initially empty $3\times 3$ array with repetition allowed but subject to the constraints given in a magic square. Even though each games ends in at most nine moves, determining the player with the winning strategy is a surprisingly challenging and elusive problem. A complete solution is presented.

February 15

* Speaker:* Brendan Rooney

*The Colin de Verdière Number of a Graph.*

**Title:*** Abstract: *This talk is the first in a series of two talks on "strong" matrix conditions. In this talk we define the >Colin de Verdière graph invariant, and show that it is a minor monotone parameter.>

The Colin de Verdière number of graph $G$ is the maximum multiplicity of $0$ as an eigenvalue over matrices compatible with the adjacency matrix of $G$. In order for $M$ to be compatible with $G$ it must (among other things) satisfy the Strong Arnold Hypothesis. Our focus will be on this property, how it can be interpreted, and how it is applied. We will also discuss the characterization of planar graphs by their Colin de Verdière number. >Our discussion follows van der Holst, Lovász, and Schrijver (*The Colin de Verdière graph parameter*, 1997).

February 22

* Speaker:* Brendan Rooney

*The Strong Spectral Property and the Strong Multiplicity Property.*

**Title:*** Abstract: *This talk is the second in a series of two talks on "strong" matrix conditions. In this talk we focus on two analogous properties to the Strong Arnold Property, and their connection to the minimum number of distinct eigenvalues of a graph.

For a graph $G$ on $n$ vertices, $\mathcal{S}(G)$ is the space of $n\times n$ symmetric matrices whose off-diagonal zero pattern exactly matches that of $A(G)$. We are interested in finding $q(G)$, the minimum number of distinct eigenvalues of a matrix $M\in\mathcal{S}(G)$. A matrix $M$ has the Strong Spectral Property (SSP) if a pair of manifolds associated with $M$ intersect transversally at $M$. We will unpick what this means, and see how it can be used to give a lower bound of $q(G)$. Our discussion follows two papers of Barrett et al., "Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph" (2016), and "The inverse eigenvalue problem of a graph: Multiplicities and minors" (2017).

March 1

* Speaker:* Shahla Nasserasr

*>A Nordhaus-Gaddum conjecture on the eigenvalues of graphs and their structures.*

**Title:*** Abstract: *For a graph $G$ on $n$ vertices, and a parameter $p$ related to $G$, a Nordhaus-Gaddum type problem is to find an upper bound and a lower bound as functions of $n$ for $p(G)+p(\overline{G})$. When $p(G)=q(G)$, the minimum number of distinct eigenvalues of $G$ (introduced in the previous talk by Dr. Rooney), it is conjectured that the upper bound is $n+2$. The conjecture is proved for many families of graphs. In this talk, we will focus on this conjecture and the families of graphs for which the conjecture is true. The talk will be out of this paper.

March 8

* Speaker:* Shahla Nasserasr

*Orthogonal matrices with a given pattern of zero entries.*

**Title:*** Abstract: *For a given graph $G$, let $S(G)$ be the set of all symmetric matrices such that the $(i,j)$ entry with $i\neq j$< is zero exactly when $ij$ is not an edge of $G$. The case when $S(G)$ contains an orthogonal matrix is especially interesting because that means $G$ has two distinct eigenvalues ($q(G)=2$). It is known that for any two connected graphs with the same number of vertices there is a symmetric orthogonal matrix compatible with their join. This statement is generalized by Levene et al. to graphs that are not necessarily connected. In this talk, we will focus on this generalization in the paper: https://arxiv.org/abs/2012.12694

March 15

* Speaker:* Darren Narayan

*Orthogonal matrices with zero diagonal.*

**Title:*** Abstract: *A presentation will be given on the paper "On orthogonal matrices with zero diagonal" by R. Bailey and R. Craigen. In this paper the authors consider real orthogonal $n\times n$ matrices whose diagonal entries are zero and off-diagonal entries nonzero. They use these results to determine the minimum number of distinct eigenvalues of matrices associated with some families of graphs, and consider the related notion of orthogonal matrices with a partially-zero diagonal.

March 29

* Speaker:* Bonnie Jacob

*Relating linkages to the inverse eigenvalue problem on a graph.*

**Title:*** Abstract: *In this talk, we will describe the results presented in Rigid linkages and partial zero forcing by Ferrero et al. (2019). We will discuss linkages and rigid linkage forcing. We will also explore how these concepts relate to matrix eigenvalue multiplicities.

April 5

* Speaker:* Louis Deaett (Quinnipiac University)

*Matroids and the minimum rank problem for matrices.*

**Title:*** Abstract: *Suppose the only information we have about a matrix is its number of rows, its number of columns, and whether each entry is zero or nonzero. What can this tell us about the rank of the matrix? This is a well-studied question in combinatorial matrix theory. In this talk, we discuss how to place this problem within the context of matroid theory, which can be thought of as a purely combinatorial abstraction of the way linear independence and rank behave over a vector space. We will revisit some known results and show how they can be better understood in terms of matroids. We will also use the connection to matroid theory to explain and improve on some existing examples, obtain a couple of new results, and highlight some new research questions opened up by this connection.

April 12

* Speaker:* Victoria McGraw

*Spanning Shallow-Light Trees.*

**Title:*** Abstract: *Shallow-Light Trees are a form of bicriteria optimization combining minimum weight spanning trees and shortest path graphs. Spanning Shallow-Light Trees are the simplest form of this optimization. However, the extensions and variations of Shallow-Light Trees are often applied to network optimization areas such as buy-at-bulk, routing topology, and data allocation problems. This presentation will provide an introduction and overview of the applications and construction of Spanning Shallow-Light Trees.

April 19

* Speaker:* Jobby Jacob

*Graphs with two distinct eigenvalues.*

**Title:*** Abstract: *We will discuss some of the results from the paper by Chen, Grimm, McMichael and Johnson, Undirected graphs of Hermitian matrices that admit only two distinct eigenvalues. In particular, we will discuss construction techniques used in this paper to study graphs where one of the two eigenvalues has multiplicity two.

April 26

* Speaker:* Noah Collins

*Extremal Graph Theory and Spectral Radii.*

**Title:*** Abstract: *One area of extremal graph theory involves looking at graphs being embedded in certain surfaces. From the paper Three Conjectures In Extremal Spectral Graph Theory

*,*the authors show the structure of planer and outer planar graphs when embedded in the plane. In particular, the authors look at the spectral radii of outer planar and planer graphs. By looking at these graphs' spectral radii and maximizing the authors are able to deduce the structure of outer planar and planar graphs when embedded in the plane.

September 18

* Speaker:* Stanisław Radziszowski

*Chromatic Vertex Folkman Numbers*

**Title:*** Abstract: *For graph $G$ and integers $a_1 \ge \cdots \ge a_r \ge 2$, we write $G \rightarrow (a_1 ,\cdots ,a_r)^v$ if and only if for every $r$-coloring of the vertex set $V(G)$ there exists a monochromatic $K_{a_i}$ in $G$ for some color $i \in \{1, \cdots, r\}$. The vertex Folkman number $F_v(a_1 ,\cdots ,a_r; s)$ is defined as the smallest integer $n$ for which there exists a $K_s$-free graph $G$ of order $n$ such that $G \rightarrow (a_1 ,\cdots ,a_r)^v$. It is well known that if $G \rightarrow (a_1 ,\cdots ,a_r)^v$ then $\chi(G) \geq m$, where $m = 1+ \sum_{i=1}^r (a_i - 1)$. In this paper, we study Folkman graphs $G$ with chromatic number $\chi(G)=m$, which leads to a new concept of chromatic Folkman numbers. We prove constructively some existential results, among others, that for all $r,s \ge 2$ there exist $K_{s+1}$-free graphs $G$ such that $G \rightarrow (s,\cdots_r,s)^v$ and $G$ has the smallest possible chromatic number $r(s-1)+1$ with respect to this property. Among others we conjecture that for every $s \ge 2$ there exists a $K_{s+1}$-free graph $G$ on $F_v(s,s;s+1)$ vertices with $\chi(G)=2s-1$ and $G\rightarrow (s,s)^v$.

The paper and slides for this talk are available at #1 here.

September 25

* Speaker:* Brendan Rooney

*Efficient Domination in Regular Graphs*

**Title:*** Abstract: *A function $f:V(G)\rightarrow\{0,\ldots,j\}$ is an efficient $(j,k)$-dominating function on $G$ if $\sum_{u\in N[v]}f(u)=k$ for all $v\in V(G)$ (here $N[v]=N(v)\cup\{v\}$ is the closed neighbourhood of $v$). Efficient $(j,k)$-domination was introduced by Rubalcaba and Slater (2007) as a generalization of perfect domination, and efficient $k$-domination. We look at efficient domination on regular graphs, applying some standard tools from linear algebra and algebraic graph theory. Using these ideas we give a partial characterization of the values $k$ for which the Hamming graphs $H(q,d)$ are efficiently $(1,k)$-dominatable. This extends the theorem of Tietavainen-van Lint-Leont’ev-Zinov’ev characterizing codes that meet the sphere packing bound.

October 2

* Speaker:* Bonnie Jacob

*Minimum Zero-Diagonal Rank and Failed Skew Zero Forcing of Graphs*

**Title:*** Abstract: *Associated with any simple graph $G$ is a family of symmetric zero-diagonal matrices with the same zero-nonzero pattern as the adjacency matrix of $G$. There is a strong connection between the ranks of these matrices and the generalized cycles that exist as subgraphs of $G$. In this talk, we characterize all connected graphs $G$ with minimum rank of $3$ or below, as well as all connected graphs with minimum rank of $n$, the order of the graph. It turns out that the minimum rank is the order of the graph if and only if $G$ has a unique generalized cycle, also known as a unique perfect $[1,2]$-factor, among other names. We present an algorithm for determining whether a graph has a unique spanning generalized cycle. We also determine the maximum zero-diagonal rank of a graph, which is also related to generalized cycles, and then show that there exist graphs $G$ for which some ranks between minimum rank and the maximum rank of $G$ cannot be realized.

Related to the notion of minimum rank is the idea of zero forcing. For minimum zero-diagonal rank, the skew zero forcing number and failed skew zero forcing number provide bounds on the rank of a graph. We present some results for the failed skew zero forcing number of a graph.

These results are based on joint work with C. Grood, L. Hogben, J. Harmse, T. J. Hunter, A. Klimas, S. McCathern, T. Ansill, J. Penzellna, and D. Saavedra. Paper.

October 9

* Speaker:* Darren Narayan

*Symmetry in Point-Block Incidence Graphs*

**Title:*** Abstract: *We present infinite families of graphs $G$ where all symmetries can be removed by fixing a single vertex. That is, mapping any vertex to itself results in the trivial automorphism. These graphs, called point-block incidence graphs, lie at the intersection of graph theory and combinatorial design theory. A point-block incidence graph is a bipartite graph $G=(P,B)$ with a set of point vertices $P=\{p_{1},p_{2},...,p_{r}\}$ and a set of blocks $B=\{B_{1},B_{2},...,B_{s}\}$ where $p_{i}\in P$ is adjacent to $B_{j}\in B\Leftrightarrow p_{i}\in B$. A vertex $v$ in a graph $G$ is fixed if it is mapped to itself under every automorphism of $G$. The fixing number of a graph $G$ is the minimum number of vertices, when fixed, fixes all of the vertices in $G$ and was introduced by Laison, Gibbons, Erwin, and Harary.

We present infinite families of graphs with a fixing number of $1$, and further fixing any vertex fixes every vertex of the graph. We also show that other point-block incidence graphs can have a high degree of symmetry and a large fixing number as they can be expressed as the disjoint union of copies of $P_{2}\times P_{n}$, $K_{3,3}$, or Möbius ladder graphs. This is joint work with Josephine Brooks, Alvaro Carbonero, Joseph Vargas, Rigoberto Flórez, and Brendan Rooney.

October 16

* Speaker:* >Kristijan Tabak (RIT Croatia)

*An Algebraic Proof That the Binary Fano Plane is Almost Rigid*

**Title:*** Abstract: *The existence of a binary $q$-analog of a Fano plane is still unknown. Kiermaier, Kurz and Wassermann proved that automorphism group of a binary $q$-analog of a Fano plane is almost trivial, it contains at most two elements. The method used there involved Kramer - Masner method together with an extensive computer search. In this paper we provide an algebraic (computer free) proof that automorphism group of a binary $q$-analog of a Fano plane contains at most two elements. We used group theory with calculations in a suitable group rings.

October 23

* Speaker:* Daniela Elizondo and Henry Fleischmann

*Efficient $(j,k)$-Domination on Chess Graphs*

**Title:*** Abstract: *Graphs defined by the legal moves of a chess piece are a classical setting for efficient domination. For a graph $G$, a function $f: V(G) \rightarrow \{0, 1, \ldots, j \}$ is an

*efficient*$(j,k)$

*-dominating function*if, for all $v \in V(G)$, $\sum_{w \in N[v]} f(w) = k$, where $N[v]$ is the closed neighborhood of $v$ (introduced by Rubalcaba and Slater, 2007).

We completely characterize efficient $(j,k)$-domination on King's graphs. This result generalizes to a construction that shows $G\boxtimes H$ is efficiently $(j,k)$-dominatable if and only if both $G$ and $H$ are. Additionally, we describe several necessary conditions for efficient $(j,k)$-domination, following our observations on Bishop's graphs.

On the torus, the Queen's and Bishop's graphs are realizable as Cayley graphs. We apply character theory to calculate the spectra of these graphs, through which we determine their efficient $(j,k)$-dominating functions.

For the standard $n\times n$ Queen's graph, we exploit an equitable partition to show computationally that, for $4 \leq n \leq 571$, efficient $(j,k)$-domination occurs only when $n = 10$. Expanding this approach, we construct an infinite class of graphs with an efficient $(j,k)$-dominating function from analogous equitable partitions.

October 30

* Speaker:* Shahla Nasserasr

*Achievable Multiplicity Partitions of a Graph*

**Title:*** Abstract: *For a graph $G$, the class of real-valued symmetric matrices whose zero-nonzero pattern of off-diagonal entries is described by the adjacencies in $G$ is denoted by $S(G)$. The inverse eigenvalue problem for the multiplicities of the eigenvalues of $G$ is to determine for which ordered list of positive integers $m_1\geq m_2\geq \cdots\geq m_k$ with $\sum_{i=1}^{k} m_i=|V(G)|$, there exists a matrix in $S(G)$ with distinct eigenvalues ${\lambda_1,\lambda_2,\cdots, \lambda_k}$ such that $\lambda_i$ has multiplicity $m_i$. A related parameter is $q(G)$, the minimum number of distinct eigenvalues of a matrix in $S(G)$. We study graphs that can achieve two distinct eigenvalues ($q(G)=2$) with given multiplicities. This is joint work with the Discrete Mathematics Research Group of Regina.

November 6

* Speaker:* Nate Cahill

*Relaxing Balanced Graph Cuts Part 2*

**Title:*** Abstract: *In April, I showed how image segmentation can be thought of as a process of partitioning an image into a small number of subsets, and how image segmentation algorithms can often be modelled as graph partitioning problems which are typically NP-hard. In this talk, we’ll focus on a particular instance of this algorithm, called “Balanced Cuts,” by describing the cost function whose minimum coincides with the optimal partitioning, and by showing how a relaxation of the discrete minimization problem yields a continuous problem that can be solved through a generalization of iteratively reweighted least squares (IRLS).

November 13

* Speaker:* John Whelan

*Mismatch Metrics, Lattice Coverings and Coordinate Choices: Adventures in the Parameter Space of Continuous Gravitational Waves>*

**Title:*** Abstract:* We describe the application of the lattice covering problem to the placement of templates in a search for continuous gravitational waves from the low-mass X-Ray binary Scorpius X-1.

Sco X-1 is a neutron star in a binary system with a low-mass companion star. It is believed to be rapidly spinning and a promising source of continuous gravitational waves. The signal received by an observatory such as LIGO, Virgo or KAGRA depends on the parameters of the system, and a search for that signal loses sensitivity if the incorrect values are used for those parameters. Several of the parameters (rotational frequency, projected orbital radius, orbital period and orbital phase) are uncertain, and one method to ensure that the signal is not missed is to perform the search at each point in a lattice covering the relevant parameter space.

The loss of signal-to-noise ratio (SNR) associated with an incorrect choice of parameters is, in a generic Taylor expansion, a quadratic function of the parameter offsets. This allows us to write the fractional loss in SNR, also known as the mismatch, as a distance using a metric on parameter space. In general, this metric will vary over the parameter space (i.e., the associated geometry will have intrinsic curvature), but we can divide the parameter space into small enough pieces that the space is approximately flat, and the metric can be assumed to be constant. In that case, there exists a transformation to Euclidean coordinates. The problem of placing templates so that the mismatch of any point in parameter space from the nearest template is no more than some maximum mismatch $\mu$ is then equivalent to the problem of covering the corresponding Euclidean space with spheres of radius $\mu$. The most efficient covering lattice in $n\le 5$ dimensions is the family $A_n^*$, which includes the hexagonal lattice $A_2^*$ and the body-centered cubic lattice $A_3^*$. For example, the density of lattice points for $A_4^*$ is a factor of $2.8$ lower than the corresponding hypercubic ($\mathbb{Z}^4$) lattice.

We use the LatticeTiling module in the LIGO Algorithms Library (lalsuite) to investigate efficient lattice coverings for the parameter space of a search for Sco X-1 using advanced LIGO data. We show how the search can be made more efficient by 1) replacing a hypercubic grid with an $A_n^*$ lattice, 2) accounting for the elliptical boundaries associated with the correlated prior uncertainties between orbital period and orbital phase, and 3) defining a "shearing" coordinate change such that a particular combination of the orbital period and orbital phase is "unresolved", and explicitly searching only in the other three dimensions of the parameter space. These improvements allow the search to be carried out using fewer computational resources. Alternatively, since the search method we use is "tunable", with a tradeoff between

computational cost and sensitivity, the more efficient lattice allows a more sensitive search to be done at the same computing cost.

This work is a collaboration between AST PhD student Katelyn Wagner and myself, with help from AST PhD student Jared Wofford.

November 20

* Speaker:* Rigoberto Florez

*Some Enumerations on Non-decreasing Dyck Paths*

**Title:*** Abstract:* A

*Dyck path*is a lattice path in the first quadrant of the $xy$-plane that starts at the origin and ends on the $x$-axis. It consists of the same number of North-East ($U$) and South-East ($D$) steps. A

*pyramid*is a subpath of the form $U^nD^n$. A

*valley*is a subpath of the form $DU$. The height of a valley is the $y$-coordinate of its lowest point. A Dyck path is called

*non-decreasing*if the heights of its valleys form a non-decreasing sequence from left to right.

In this talk, we count several aspects of non-decreasing Dyck paths. We count, for example, the number and weight of pyramids and numbers of primitive paths. In the end of the talk we introduce the concept of symmetric pyramids and count them. Throughout the talk, we give connections (bijective relations) between non-decreasing Dyck paths with other objects of the combinatorics. Some examples are, words, trees, polyominoes. This is a joint work with Eva Czabarka, José L. Ramírez, and Leandro Junes.

February 12

* Speaker:* Brendan Rooney

*An Introduction to Cayley Graphs*

**Title:*** Abstract: *Cayley graphs are graphs constructed from groups. We show how questions about graph symmetry naturally lead to Cayley graphs. We also survey some classical results about Cayley graphs.

February 19

* Speaker:* Brendan Rooney

*More Cayley Graphs*

**Title:*** Abstract: *We continue our discussion of Cayley graphs, shifting our focus to structural questions. We will look at cliques, neighborhoods, colorings, and more!

February 26

* Speaker:* Bonnie Jacob

*Construction of Graphs That Swap Distance-One and Distance-Two Pairs of Vertices*

**Title:*** Abstract:* The

*distance*$d_G(u,v)$ between vertices $u$ and $v$ in a graph $G$ with vertex set $V(G)$ is the number of edges on a shortest path from $u$ to $v$. From a graph $G$, suppose we want to construct a new graph $H$ that has the same vertex set of $G$, but $d_{H}(u,v) = 1$ if and only if $d_G(u,v)=2$, and $d_{H}(u,v) = 2$ if and only if $d_G(u,v)=1$. For most choices of $G$, we would be out of luck. However, if $G=C_5$, for example, we can find such a graph $H$. In fact, $H=G=C_5$.

In this talk, we will discuss the motivation for considering this property and the conditions under which it holds.

March 18

* Speaker:* Brendan Rooney

*Hardness of Finding Maximum Cliques in Cayley Graphs*

**Title:*** Abstract:* Computing the clique number of a general graph is well-known to be NP-Hard. Codenotti et al. [Bruno Codenotti, Ivan Gerace, and Sebastiano Vigna. Hardness results and spectral techniques for combinatorial problems on circulant graphs. Linear Algebra Appl., 285(1-3): 123--142, 1998] showed that computing clique number and chromatic number are still NP-Hard problems for the class of circulant graphs. This result demonstrates that the assumption of vertex transitivity does not make computing clique numbers or chromatic numbers easier. It also raises the question of whether there are classes of Cayley graphs for which these problems are easy to solve.

We outline proof that computing clique number is NP-Hard for the class of Cayley graphs for the groups $G^n$, where $G$ is any fixed finite group. In particular, this shows that these problems are NP-Hard for cubelike graphs. Our method brings together: a construction for embedding graphs in Cayley graphs; quotients of groups; quotients of graphs; and, Goppa codes.

March 25

* Speaker:* Nishant Malik

*Networks based analysis of regional and global climate systems*

**Title:*** Abstract:* Analysis of large-scale spatiotemporal datasets of various regional and global climate phenomena is a challenging task, a recently developed innovative way to analyze these datasets is employing methods of network science. These methods are especially useful in situations where the underlying motivation for the analysis is identifying connectivity patterns between various dynamical features of the climate system. For example, using these methods, we can answer questions such as the role of ENSO (El Niño–Southern Oscillation) in global rainfall patterns.

In this talk, I will illustrate the effectiveness of network science-based climate data analysis by applying it to rainfall patterns over South Asia and East Asia. For monsoon over South Asia, I will show that we can identify various spatial patterns that precede extreme rainfall, these patterns can not only be used as early warning signs for extreme rainfall events but can also provide insights into various dynamical features of monsoon. Furthermore, for the East Asian region, using the community detection algorithm, we delineate regions of coherent rainfall during different seasons. Time permitting, I will show some of our recent work on predicting monsoon and ENSO using network analysis.

April 1

* Speaker:* Manuel Lopez

*Finding polynomials for functions modulo a prime with a little help from linear algebra>*

**Title:*** Abstract: *Every function $f:\mathbb{Z}/p\mathbb{Z}\rightarrow\mathbb{Z}/p\mathbb{Z}$, where $p$ is a prime, can be expressed as a polynomial. There wasn't any easily computable approach I could find, so I worked out one approach. I will also motivate why such a computational tool is useful for number-theoretic purposes.

April 8

* Speaker:* Darren Nararyan

*Graph theory metrics for analyzing functional MRI data and brain connectivity*

**Title:*** Abstract: * In recent years metrics from the theory of graphs and networks have emerged as powerful means for analyzing functional MRI data. These include an array of properties found commonly applied to the realm of transportation and social networks, such as the characteristic (average) path length, the clustering coefficient, as well as global and local efficiency. We have also focused on the concept of betweenness centrality, which measures the frequency in which shortest paths between regions pass through a particular region or connection. The talk will include an overview of these methods and show how they can be applied to analyze brain connectivity.

April 15

* Speaker:* Matthew Coppenbarger

*An Introduction to Combinatorial Game Theory*

**Title:*** Abstract:* Combinatorial Game Theory is based on a simple and intuitive recursive definition of games, which yields a very rich algebraic structure: games can be added and subtracted in a very natural way, forming an abelian group with a partial ordering. The talk will also introduce some of the fundamental games that motivate the theory.

April 22

* Speaker:* Nathan Cahill

*Relaxing Balanced Graph Cuts*

**Title:*** Abstract:* In the computer vision community, a huge focus of research over the last generation has been on analyzing and understanding images. One common “low-level” image analysis task, image segmentation, aims to partition an image into a small number of subsets that can be input into subsequent high-level image understanding tasks. In this talk, we’ll explore how image segmentation algorithms can often be modeled as graph partitioning problems which are typically NP-hard, and then we’ll discuss how some of these problems can be relaxed to generate heuristics that can be solved rapidly.

April 29

* Speaker:* Stanislaw Radziszowski

*Bounds on Shannon Capacity and Ramsey Numbers from Product of Graphs*

**Title:*** Abstract:* We study the Shannon capacity of channels in the context of Ramsey numbers. We overview some of the results on channel capacity, and how Ramsey-type constructions may enhance them. A new lower bound for a special type of multicolor Ramsey numbers is presented, which implies that the supremum of the Shannon capacity over all graphs with independence 2 cannot be achieved by any finite graph power. This generalizes to graphs with any bounded independence number.

The paper and slides are available at #33 here.

September 25

** Speaker:** Brendan Rooney

**Hedetniemi's Conjecture: Past, Present, and Future**

*Title:*** Abstract:** In 1966 Stephen Hedetniemi conjectured that the chromatic number of the direct product of graphs $G$ and $H$ is equal to the minimum of the chromatic numbers of $G$ and $H$. In the 53 years it has stood open, this conjecture has inspired a great deal of research on graph colourings and homomorphisms. Very recently (May of 2019!), Yaroslav Shitov published a construction on the arXiv giving counterexamples to Hedetniemi's Conjecture. I will give an introduction to this conjecture, some evidence for its validity, an account of Shitov's construction, and some results that suggest further directions for research.

October 9

* Speaker:* Stanisław Radziszowski

*On a Diagonal Conjecture for Classical Ramsey Numbers*

**Title:**** Abstract:** Let $R(k_1,\ldots,k_r)$ denote the classical $r$-color Ramsey number for integers $k_i\geq 2$. The Diagonal Conjecture (DC) for classical Ramsey numbers poses that if $k_1,\ldots,k_r$ are integers no smaller than $3$ and $k_{r-1}\leq k_r$, then $R(k_1,\ldots,k_{r-2},k_{r-1}-1,k_r+1)\leq R(k_1,\ldots,k_r)$. We obtain some implications of this conjecture, present evidence for its validity, and discuss related problems.

Let $R_r(k)$ stand for the $r$-color Ramsey number $R(k,\ldots,k)$. It is known that $\lim_{r\rightarrow\infty}R_r(3)^{1/r}$ exists, either finite or infinite, the latter conjectured by Erdős. This limit is related to the Shannon capacity of complements of $K_3$-free graphs. We prove that if (DC) holds, and $\lim_{r\rightarrow\infty}R_r(3)^{1/r}$ is finite, then $\lim_{r\rightarrow\infty}R_r(k)^{1/r}$ is finite for every integer $k\geq 3$.

* Paper:* arxiv.org/abs/1810.11386.

*ram19a.pdf.*

**Slides:**October 23

* Speaker: *Bonnie Jacob

*Results and Remaining Questions About Failed Zero Forcing on Directed Graphs*

**Title:*** Abstract:* Let $D$ be a simple digraph (directed graph) with vertex set $V(D)$ and arc set $A(D)$ where $n=|V(D)|$, and each arc is an ordered pair of distinct vertices. If $(v,u) \in A(D)$, then $u$ is considered an

*out-neighbor*of $v$ in $D$. Initially, we designate each vertex to be either filled or empty. Then, the following color change rule (CCR) is applied: if a filled vertex $v$ has exactly one empty out-neighbor $u$, then $u$ will be filled. The process continues until either all of the vertices in $V(D)$ are filled, or the CCR restricts the remaining empty vertices from being filled. If all vertices in $V(D)$ are eventually filled, then the initial set is called a

*zero forcing set*(ZFS); if not, it is a

*failed zero forcing set*(FZFS).

In this talk, we define the *failed zero forcing number* $\operatorname{F}(D)$ on a digraph, which is the maximum cardinality of any FZFS. The *zero forcing number*, $\operatorname{Z}(D)$, is the minimum cardinality of any ZFS. We characterize oriented graphs that have $\operatorname{F}(D)<\operatorname{Z}(D)$ and present a list of digraphs with the same property. We also characterize digraphs with high and low values of $\operatorname{F}(D)$. We also discuss some open questions that remain about failed zero forcing on directed graphs.

This is based partially on work during the 2018 REU together with Alyssa Adams of Youngstown State University.

November 6

* Speaker:* Darren Narayan

*The Asymmetric Index of a Graph*

**Title:*** Abstract:* A graph $G$ is

*asymmetric*if its automorphism group is trivial. Asymmetric graphs were introduced by Erdős and Rényi (1963). They suggested the problem of starting with an asymmetric graph and removing some number $r$ of edges and/or adding some number $s$ of edges so that the resulting graph is non-asymmetric. Erdős and Rényi defined the

*degree of asymmetry*of a graph to be the minimum value of $r+s$. In this paper, we define another property that measures how close a given non-asymmetric graph is to being asymmetric. We define the

*asymmetric index*of a graph $G$, denoted $ai(G)$, to be the minimum of $r+s$ so that the resulting graph $G$ is asymmetric.

We investigate the asymmetric index of both connected and disconnected graphs. We prove that for any non-negative integer $k$, there exists a graph $G$ where $ai(G)=k$. We show that the asymmetric index of a cycle with at least six vertices is two, and provide a complete characterization of all possible pairs of edges that can be added to a cycle to create an asymmetric graph. In addition we determine the asymmetric index of paths, certain circulant graphs, Cartesian products involving paths and cycles, and bounds for complete graphs, and complete bipartite graphs.

November 20

* Speaker:* Jobby Jacob

*On the Characterization of Graphs Based on Their Rank Numbers*

**Title:*** Abstract: *For a graph $G$, an onto function $f$ from the vertex set of $G$ to $\{1,2,\ldots,k\}$ is a $k$-ranking if $f(u)=f(v)$ implies that every $uv$-path contains a vertex $x$ such that $f(x)>f(u)=f(v)$. The rank number of a graph $G$ is the minimum value of $k$ such that $G$ has a $k$-ranking.

We will discuss the rankings of some classes of graphs, and in particular, we will discuss a recent result on characterizing graphs with large rank numbers.

December 4

* Speaker:* Matthew Coppenbarger

*Iterations of the Sisyphus Function*

**Title:*** Abstract:* The Sisyphus function is defined; and we determine the smallest non-negative integer $n$ requiring a specified number of iterations of the function that must be applied to $n$ until the sequence generated by the iterations of this function becomes stable or cycles.