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.

Mathematics Subject Classifications: 05C35, 05C50

Keywords: Spectral Turán number, even-cycle problem, Brualdi-Solheid problem