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In this paper, we study the maximum adjacency spectral radii of graphs of large order that do not contain an even cycle of given length. For n > k, let S n,k be the join of a clique on k vertices with an independent set of n − k vertices and denote by S + n,k the graph obtained from S n,k by adding one edge. In 2010, Nikiforov conjectured that for n large enough, the C 2k+2 -free graph of maximum spectral radius is S + n,k and that the {C 2k+1 , C 2k+2 }-free graph of maximum spectral radius is S n,k . We solve this two-part conjecture.

The odd wheel W 2k+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 W 2k+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 W 2k+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.