A spectral condition for odd cycles in non-bipartite graphs (Q2238862)

For a graph \(G\), the spectral radius \(\rho(G)\) of \(G\) is the largest eigenvalue modulus of the adjacency matrix of \(G\). \textit{H. Lin} et al. [Comb. Probab. Comput. 30, No. 2, 258--270 (2020; Zbl 1466.05121)] posed the question: which class of graphs attain the maximum spectral radius among all non-bipartite graphs of order \(n\) and odd girth at least \(2k+3\)? The present authors answer this question: the class in question consists of exactly one graph, namely the graph obtained by replacing an edge of \(K_{\lceil\frac{n-2k+1}{2}\rceil,\lfloor\frac{n-2k+1}{2}\rfloor}\) by the path \(P_{2k+1}\).