Spectral Conditions for a Graph to be Hamilton-Connected

Abstract

In this paper we establish some spectral conditions for a graph to be Hamilton-connected in terms of the spectral radius of the adjacency matrix or the signless Laplacian of the graph or its complement. For the existence of Hamiltonian paths or cycles in a graph, we also give a sufficient condition by the signless Laplacian spectral radius.

Sufficient conditions for Hamilton-connected graphs in terms of (signless Laplacian) spectral radius

“…(iii) If q(G) > 2n − 4 + 2 n−1 , then G is Hamilton-connected unless G = K n−1 + e + e ′ . Recently, Zhou and Wang [31] gave some spectral sufficient conditions on spectral radius and signless Laplacian spectral radius for a graph to be Hamilton-connected, which extended the result of Yu and Fan [26] in some sence. Theorem 1.7.…”

“…Lemma 2.7. ( [8,26]) Let G be a connected graph on n vertices and m edges. Then q(G) ≤ 2m n−1 + n − 2.…”

“…Fiedler and Nikiforov [9] firstly gave sufficient conditions in terms of the spectral radius of a graph or its complement for the existence of Hamilton cycles. This work motivated further research, one may refer to [1,18,19,23,26,27,28,30]. Recently, by imposing the minimum degree of a graph as a new parameter, Li and Ning [14,15] extended some the results in [9,18,23].…”

“…In [26], Yu and Fan have established sufficient conditions for a graph to be Hamiltonconnected in terms of the spectral radius and signless Laplacian spectral radius. Write G = K n−1 + e + e ′ for K n−1 together with a vertex joining two vertices of K n−1 by the edges e, e ′ , respectively.…”

Distance signless Laplacian spectral radius and Hamiltonian properties of graphs

“…In 2003, Krivelevich and Sudakov first proposed a sufficient condition on the spectrum of the adjacency matrix for a regular graph to be Hamiltonian, where the graphs satisfying the given condition are pseudo-random. Some other spectral conditions for Hamilton cycles and paths in graphs have been given in [1,2,4,5,6,8,11,9,13,14]. In this paper, we mainly consider the relationship between distance signless Laplacian spectral radius and the Hamiltonian properties of graphs.…”