The spectral radius of graphs with no odd wheels

Abstract

The odd wheel W2k+1 is the graph formed by joining a vertex to a cycle of length 2k. In this paper, we investigate the largest value of the spectral radius of the adjacency matrix of an n-vertex graph that does not contain W2k+1. We determine the structure of the spectral extremal graphs for all k≥2,k⁄∈{4,5}. When k=2, we show that these spectral extremal graphs are among the Turán-extremal graphs on n vertices that do not contain W2k+1 and have the maximum number of edges, but when k≥9, we show that the family of spectral extremal graphs and the family of Turán-extremal graphs are disjoint.