An Aα-spectral Erdos-Posa theorem

摘要

Given a graph G and a real number alpha is an element of[0, 1], Nikiforov (2017) proposed the A(alpha)-matrix of Gas A(alpha)(G) = alpha D(G) +(1 - alpha) A( G), where A(G) and D( G) are the adjacency matrix and the degree diagonal matrix of G, respectively. The largest eigenvalue of A(alpha)(G), written as lambda(alpha)(G), is called the A alpha-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 S-n,S- 2k-1 be the join of a clique on 2k - 1vertices with an independent set of n - (2k - 1) vertices. The famous Erdos-Posa theorem shows that for k >= 2 and n >= 24k, every n-vertex graph G with at least (2k - 1)(n - k) edges contains kindependent cycles, unless G congruent to S-n,S- 2k-1. In this paper, we consider an A(alpha)-spectral version of this theorem. We show that for fixed k >= 1, 0 < alpha < 1and n >= 104k(3)/alpha(a)(1- alpha), if an n-vertex graph Gsatisfies lambda(alpha)(G) >= lambda(alpha)(S-n,S- 2k-1), then it contains kindependent cycles, unless G congruent to Sn, 2k-1. This extends the result of Zhai and Liu (2022), in which they obtained the adjacency spectral version of the Erd.os-Posa theorem.

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