A sharp upper bound on the spectral radius of C5-free /C6-free graphs with given size

Let S-n,S-2 be the graph obtained by joining each vertex of K-2 to n - 2 isolated vertices, and let S-n,2(-) be the graph obtained from S-n,S-2 by deleting an edge incident to a vertex of degree two. Recently, Zhai, Lin and Shu [20] showed that rho(G) <= 1+root 4m-3/2 for any C-5-free graph of size m >= 8 or C-6-free graph of size m >= 22, with equality if and only if G congruent to S-m+3/2,S-2 (possibly, with some isolated vertices). However, this bound is sharp only for odd m. Motivated by this, we want to obtain a sharp upper bound of rho(G) for C-5-free or C-6-free graphs with medges. In this paper, we prove that if Gis a C-5-free graph of even size m >= 14 or C-6-free graph of even size m >= 74, and G contains no isolated vertices, then rho(G) <= (rho) over tilde (m), with equality if and only if G congruent to S-m+4/2,2(-), where (rho) over tilde (m) is the largest root of x(4) - mx(2) - (m - 2)x + ( m/2 - 1) = 0. (c) 2022 Elsevier Inc. All rights reserved.