  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.
