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

Abstract

Let Sn,2 be the graph obtained by joining each vertex of K2 to n−2 isolated vertices, and let Sn,2− be the graph obtained from Sn,2 by deleting an edge incident to a vertex of degree two. Recently, Zhai, Lin and Shu [20] showed that ρ(G)≤1+4m−32 for any C5-free graph of size m≥8 or C6-free graph of size m≥22, with equality if and only if G≅Sm+32,2 (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 ρ(G) for C5-free or C6-free graphs with m edges. In this paper, we prove that if G is a C5-free graph of even size m≥14 or C6-free graph of even size m≥74, and G contains no isolated vertices, then ρ(G)≤ρ˜(m), with equality if and only if G≅Sm+42,2−, where ρ˜(m) is the largest root of x4−mx2−(m−2)x+(m2−1)=0.