An Aα-spectral Erdős-Pósa theorem

Abstract

Given a graph G and a real number α∈[0,1], Nikiforov (2017) proposed the Aα-matrix of G as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G, respectively. The largest eigenvalue of Aα(G), written as λα(G), is called the Aα-index of G. A set of cycles in a graph G is called independent if no two cycles in it have a common vertex in G. For n>2k−1, let Sn,2k−1 be the join of a clique on 2k−1 vertices with an independent set of n−(2k−1) vertices. The famous Erdős-Pósa theorem shows that for k≥2 and n≥24k, every n-vertex graph G with at least (2k−1)(n−k) edges contains k independent cycles, unless G≅Sn,2k−1. In this paper, we consider an Aα-spectral version of this theorem. We show that for fixed k≥1,0<α<1 and n≥104k3α4(1−α), if an n-vertex graph G satisfies λα(G)≥λα(Sn,2k−1), then it contains k independent cycles, unless G≅Sn,2k−1. This extends the result of Zhai and Liu (2022), in which they obtained the adjacency spectral version of the Erdős-Pósa theorem.