The maximum spectral radius of {C3,C5}-free graphs of given size

Abstract

In this paper we consider the extremal problem on adjacency spectral radius of {C3,C5}-free graphs. Assume that G is a graph with m edges having no isolated vertices, and let λ be the spectral radius of its adjacency matrix. Firstly, by using the method of characterizing {C3,C5}-free non-bipartite graphs whose second largest eigenvalue is less than 54, we show that, if G is a {C3,C5}-free non-bipartite graph of size m, thenλ⩽∑u∈VGd(u)2−m+4f−54. Equality holds if and only if G≅C7, where d(u) is the degree of vertex u and f denotes the number of 4-cycles in G. Secondly, we show that, if G is a {C3,C5}-free non-bipartite graph of odd size m, then λ⩽θ(m) with equality if and only if G≅RK2,m−32, where θ(m) is the largest root ofx4−x3−(m−3)x2+(m−4)x+m−5=0 and RK2,m−32 is obtained by replacing an edge of the complete bipartite graph K2,m−32 with P5.