# Search Research Symposium Abstracts

Page 1 of 1

## Sums of Riordan Arrays

Riordan arrays are lower triangular matrices extending infinitely to the right and downward, whose columns encode generating functions. Each Riordan array is associated with two sequences, the $A$- and $Z$- sequences. While Riordan arrays are closed under multiplication, they are not closed under addition in general. We will discuss the conditions under [Read More...]

**Presenter:**Caroline Bang, Iowa State University

**Authors:**Caroline Bang, Matias Von Bell, Eric Culver, Jessica Dickson, Stoyan Simitrov, Rachel Perrier, Sheila Sundaram.

**Symposium Year:**2022

**Session:**Women from the Graduate Research Workshop in Combinatorics

**Presentation Time:**June 18, 2022; 3:20 pm

## Long cycles in Balanced Tripartite Graphs

In this talk, we will survey the relevant literature, namely degree and edge conditions for Hamiltonicity and long cycles in graphs, including bipartite and $k$-partite results. We will then prove that if $G$ is a balanced tripartite graph on $3n$ vertices, $G$ must contain a cycle of length at least $3n-1$, provided that $e(G) \geq 3n^2-4n+5$ and $n\geq [Read More...]

**Presenter:**ZHANAR BERIKKYZY, Fairfield University

**Authors:**G. Araujo-Pardo, Zh. Berikkyzy, J. Faudree, K. Hogenson, R. Kirsch, L. Lesniak, and J. McDonald

**Symposium Year:**2022

**Session:**Women from the Graduate Research Workshop in Combinatorics

**Presentation Time:**June 18, 2022; 3:45 pm

## Lower Bounds on the Erdős-Gyárfás Problem

Given positive integers $p$ and $q$, a $(p,q)$-coloring of the complete graph $K_n$ is an edge-coloring in which every $p$-clique receives at least $q$ colors. Erdős and Shelah posed the question of determining $f(n,p,q)$, the minimum number of colors needed for a $(p,q)$-coloring of $K_n$. Recently, Pohoata and Sheffer introduced the color energy technique [Read More...]

**Presenter:**Emily Heath, Iowa State University

**Authors:**József Balogh, Sean English, Emily Heath, Robert Krueger

**Symposium Year:**2022

**Session:**Women from the Graduate Research Workshop in Combinatorics

**Presentation Time:**June 18, 2022; 4:35 pm

## Narayana numbers from the geometry of Shi arrangements*

The Catalan numbers have many interpretations (counting Dyck paths, triangulations, etc). One particularly well-known refinement comes from the Narayana numbers, which can be interpreted as counting Dyck paths with a fixed number of peaks. In this talk we introduce a geometric interpretation of the Narayana numbers, via the characteristic polynomial of the [Read More...]

**Presenter:**Galen Dorpalen-Barry, Ruhr-University Bochum

**Authors:**Galen Dorpalen-Barry, Christian Stump

**Symposium Year:**2022

**Session:**Women from the Graduate Research Workshop in Combinatorics

**Presentation Time:**June 19, 2022; 8:50 am

## Graph irregularities of maximal k-degenerate graphs

A graph is maximal $k$-degenerate if every subgraph has a vertex of degree at most $k$, and the property does not hold if any new edge is added to the graph. A well-known subclass of maximal $k$-degenerate graphs is the $k$-trees. Albertson irregularity (resp., sigma irregularity) of a graph is the summation of edge imbalances (resp., squares of edge [Read More...]

**Presenter:**Zhongyuan Che, Penn State University, Beaver Campus

**Authors:**Alan Bickle and Zhongyuan Che

**Symposium Year:**2022

**Session:**Women from the Graduate Research Workshop in Combinatorics

**Presentation Time:**June 19, 2022; 9:15 am

## Maximizing Edges in the Lights Out Game*

Lights Out was a game by Tiger Electronics released in 1995. The game consisted of a 5x5 grid of buttons that could be on or off. Some of the buttons start as on, and the goal is to turn all the lights off by pushing buttons. When a button is pushed, that light and its 4 neighbors on the grid are "toggled" meaning, on switches to off and off [Read More...]

**Presenter:**Lauren Keough, Grand Valley State University

**Authors:**Lauren Keough, Darren Parker

**Symposium Year:**2022

**Session:**Women from the Graduate Research Workshop in Combinatorics

**Presentation Time:**June 19, 2022; 9:40 am

## Cospectral constructions using cousin vertices

Once a graph is associated with a matrix according to some rule, we can find the spectrum of a graph with respect to a matrix. Two graphs are cospectral if they share a spectrum. The spectrum holds some structural information about the graph. Constructions of cospectral graphs help us establish patterns about graph properties not preserved by the spectrum. [Read More...]

**Presenter:**Kate J Lorenzen, Linfield University

**Symposium Year:**2022

**Session:**Women from the Graduate Research Workshop in Combinatorics

**Presentation Time:**June 19, 2022; 10:05 am

## Cycle-Free Saturation Games*

It's all fun and games until someone loses a subgraph! Originally proposed by Hajnal, the saturation game involves two players, Mini and Maxi, who take turns adding single edges to a graph with $n$ vertices and no edges. Each player has their own goal to minimize or maximize, respectively, the number of edges in the graph when the game ends. Throughout [Read More...]

**Presenter:**Erin Meger, Wilfrid Laurier University

**Authors:**Sean English, Tomáš Masařík Grace McCourt, Erin Meger, Michael S. Ross, Sam Spiro

**Symposium Year:**2022

**Session:**Women from the Graduate Research Workshop in Combinatorics

**Presentation Time:**June 19, 2022; 10:30 am

Page 1 of 1