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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₍, ₊\) be the join of a clique on \ (k\) vertices with an independent set of \ (n-k\) vertices and denote by \ (S₍, ₊^+\) the graph obtained from \ (S₍, ₊\) by adding one edge. In 2010, Nikiforov conjectured that for \ (n\) large enough, the \ (C₂₊+₂\) -free graph of maximum spectral radius is \ (S₍, ₊^+\) and that the \ (\C₂₊+₁, C₂₊+₂\\) -free graph of maximum spectral radius is \ (S₍, ₊\). We solve this two-part conjecture. Mathematics Subject Classifications: 05C35, 05C50Keywords: Spectral Turán number, even-cycle problem, Brualdi-Solheid problem
Cioab et al. (Mon,) studied this question.