Algebraic Graph Theory: The spectral radius of graphs with no odd wheels

Talk by Dheer Noal Desai.

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 Turán-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).