Adjacency eigenvalues of graphs without short odd cycles

Abstract

It is well known that spectral Turán type problem is one of the most classical problems in graph theory. In this paper, we consider the spectral Turán type problem. Let G be a graph and let G be a set of graphs, we say G is G-free if G does not contain any element of G as a subgraph. Denote by λ1 and λ2 the largest and the second largest eigenvalues of the adjacency matrix A(G) of G, respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. Firstly, an upper bound on λ12k+λ22k of n-vertex {C3,C5,…,C2k+1}-free graphs is established, where k is a positive integer. All the corresponding extremal graphs are identified. Secondly, a sufficient condition for non-bipartite graphs containing an odd cycle of length at most 2k+1 in terms of its spectral radius is given. At last, we characterize the unique graph having the maximum spectral radius among the set of n-vertex non-bipartite graphs with odd girth at least 2k+3, which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and triangles in graphs, Combin. Probab. Comput. 30 (2) (2021) 258-270].