### Dheer Noal Desai

Affiliation: University  of Delaware, USA

Title: Spectral Turan Problems on trees and even cycles

Abstract: In this talk, we discuss some recent progress with the spectral analogue of a few Tur\’an problems: Instead of maximizing the number of edges, our objective is to maximize the spectral radius of the adjacency matrices of graphs not containing some subgraphs.

We discuss an overview comparing extremal graphs for both kinds of problems. The asymptotics of the Tur\’an numbers for graphs with chromatic number at least three is given by a celebrated theorem of Erd\H{o}s, Stone and Simonovits, and a similar result holds for the spectral Tur\’an numbers. However, the asymptotics are not known for several basic bipartite graphs.
We discuss a method that was initially used to obtain spectral Tur\’an results when the forbidden graphs had chromatic number more than two, and has recently been used to find spectral extremal graphs for even cycles and trees.

You can watch the talk here, and the slides are available here.

This talk was given on July 4th, 2022

Title: The spectral radius of graphs with no odd wheels

Abstract: The odd wheel $W_{2k+1}$ is the graph formed by joining a vertex to a cycle of length $2k$. In this talk, we will investigate the largest value of the spectral radius of the adjacency matrix of an $n$-vertex graph that does not contain $W_{2k+1}$. We determine the structure of the spectral extremal graphs for all $k \geq 2$, k not equal to 4 and 5. When $k=2$, we show that these spectral extremal graphs are among the Tur\’an-extremal graphs on $n$ vertices that do not contain $W_{2k+1}$ and have the maximum number of edges, but when $k \geq 9$, we show that the family of spectral extremal graphs and the family of Turán-extremal graphs are disjoint.

We will give an overview of similar results and describe a method that may help us find new ones. This is joint work with Sebastian Cioaba (University of Delaware) and Michael Tait (Villanova University).

You can watch the talk here, and the slides are available here.

This talk was given on June 7th, 2021