Spectral radius and Hamiltonicity of graphs with large minimum degree

Abstract

This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting λ (G) denote the spectral radius of the adjacency matrix of a graph G, the main results of the paper are:(1) Let k ≥ 1, n ≥ k 3 /2 + k + 4, and let G be a graph of order n, with minimum degree(2) Let k ≥ 1, n ≥ k 3 /2 + k 2 /2 + k + 5, and let G be a graph of order n, with minimum degreethen G has a Hamiltonian path, unlessIn addition, it is shown that in the above statements, the bounds on n are tight within an additive term not exceeding 2. cycles in G. This work motivated further research, as could be seen, e.g., in [1,10,13,14,15,16,17,21].The aim of this paper is to extend some recent results by Benediktovich [1], Li and Ning [14], and Ning and Ge [17]. To state those results we need to introduce three families of extremal graphs, which are denoted by L k (n) , M k (n) and N k (n).Write K s and K s for the complete and the edgeless graphs of order s. Given graphs G and H, write G ∨ H for their join and G + H for their disjoint union.The graphs L k (n) . For any k ≥ 1 and n ≥ 2k + 1, letThus, the graph L k (n) consists of a K n−k and a K k+1 sharing a single vertex.The graphs M k (n) . For any k ≥ 1 and n ≥ 2k + 1, letThus, the graph M k (n) consists of a K n−k and a set of k independent vertices all joined to some k vertices of the K n−k .The graphs N k (n) . For any k ≥ 1 and n ≥ 2k + 1, letThus, the graph N k (n) consists of a K n−k−1 and a set of k + 1 independent vertices all joined to some k vertices of the K n−k−1 .Note that for any admissible k and n, the graphs L k (n) and M k (n) contain no Hamiltonian cycle and N k (n) contains no Hamiltonian path, whereas the minimum degree of each of them is exactly k.The graphs M k (n) and N k (n) were used by Erdős [7] as extremal graphs in his results on Hamiltonicity of graphs with large minimum degree. Moreover, recently Li and Ning [14] showed that M k (n) and N k (n) are relevant also for some spectral analogs of Erdős's results: Ning [14]) Let k ≥ 0 and G be a graph of order n, with minimum degree δ (G) ≥ k.(1) If n ≥ max 6k + 10, (k 2 + 7k + 8)/2 and λ (G) ≥ λ (N k (n)) , then G has a Hamiltonian path, unless G = N k (n);(2) If k ≥ 1, n ≥ max 6k + 5, (k 2 + 6k + 4)/2 , and λ (G) ≥ λ (M k (n)) , then G has a Hamiltonian cycle, unless G = M k (n) .

Signless Laplacian spectral radius and Hamiltonicity of graphs with large minimum degree

“…We note for any admissible k and n, graphs L k (n) and M k (n) are not Hamiltonian. Strengthening a result by Li and Ning [8], Nikiforov [10] proved the following theorem.…”

“…The main work is to prove a lower bond on the difference between the largest entry and the smallest entry (of a particular subset of vertices) of the principal eigenvector of a family of graphs. We mention here that the idea in the proof of Lemma 2 can be used to give an alternative proof for Theorem 6 in [10]. We next construct a graph that is a variant of M k (n).…”

“…Then we strengthen a result by Li and Ning [8] for balanced bipartite graphs. Before we state our theorems, we briefly recall results from [10]. We need to introduce two families of graphs.…”

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

“…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]. Now, their results were improved by Nikiforov [22], Chen et al [5], Ge et al [10] and Li et al [17], in some sense.…”

Some new sufficient conditions for 2p-Hamilton-biconnectedness of graphs

“…Recently, Li and Ning [10] gave the spectral analogue of Moon-Moser's theorem [13]. For more results, readers are referred to [1,3,9,11,12,14,16].…”