Spectral Radius, Edge-Disjoint Cycles and Cycles of the Same Length

Abstract

 In this paper, we provide spectral conditions for the existence of two edge-disjoint cycles and two cycles of the same length in a graph, which can be viewed as the spectral analogues of Erdős and Posa's condition and Erdős' classic problem about the maximum number of edges of a graph without two edge-disjoint cycles and two cycles of the same length, respectively. Furthermore, we give a spectral condition to guarantee the existence of $k$ edge-disjoint triangles in a graph.