The spectral radius of graphs with no odd wheels

Abstract The odd wheel W 2 k + 1 is the graph formed by joining a vertex to a cycle of length 2 k . 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 W 2 k + 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 Turan-extremal graphs on n vertices that do not contain W 2 k + 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 Turan-extremal graphs are disjoint.