Spectral conditions and Hamiltonicity of a balanced bipartite graph with large minimum degree☆
Introduction
Let be a simple graph of order n with vertex set and edge set E(G). Denote by the number of edges of the graph G. Let NG(v) be the set of vertices which are adjacent to v in G. The degree of v is denoted by (or simply d(v)), the minimum degree of G is denoted by δ(G). A regular graph is one whose vertices all have the same degrees. Let be a bipartite graph with two part sets X, Y. If is called a balanced bipartite graph. If is called a nearly balanced bipartite graph. The bipartite semi-regular graph is a bipartite graph for which the vertices in the same part have the same degrees. For two disjoint graphs G1 and G2, the union of G1 and G2, denoted by is defined as and ; and the join of G1 and G2, denoted by G1∨G2, is defined as and . Denote Kn the complete graph on n vertices, On the empty graph on n vertices (without edges), the complete bipartite graph with two parts having n, m vertices, respectively.
The adjacency matrix of G is defined to be a matrix of order n, where if vi is adjacent to vj, and otherwise. The degree matrix of G is denoted by . The matrix is called the signless Laplacian matrix of G. Obviously, A(G), Q(G) are real symmetric matrix. So their eigenvalues are real number and can be ordered. The largest eigenvalue of A(G), denoted by μ(G), and the corresponding eigenvectors (whose all components are positive number) for G are called the spectral radius and the Perron vector of G, respectively. The largest eigenvalue of Q(G) is called the signless Laplacian spectral radius of G.
A Hamiltonian cycle of the graph G is a cycle which contains all vertex in G, and a Hamiltonian path of G is a path which contains all vertex in G. A graph G is said to be Hamiltonian if it contains a Hamiltonian cycle. A graph G is said to be traceable if it contains a Hamiltonian path. The problem of deciding whether a graph is Hamiltonian or traceable is one of the most difficult classical problems in graph theory. Indeed, it is NP-complete.
Recently, the spectral theory of graphs has been applied to this problem. Up to now, there are some references on the spectral conditions for a graph to be traceable or Hamiltonian. We refer readers to see [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. Particularly, Li and Ning [10], [11], Nikiforov [12] and Yu et al. [15] study the graphs with large minimum degree. Li and Ning [10], [11] present some spectral radius conditions for a balanced bipartite graph to be traceable, and some (signless Laplacian) spectral radius conditions for a simple graph and a balanced bipartite graph to be traceable or Hamiltonian, respectively. Nikiforov [12] gives some spectral radius conditions for a simple graph to be traceable and Hamiltonian, respectively. Yu et al. [15] present some (signless Laplacian) spectral radius conditions for a nearly balance bipartite graph to be traceable. Motivated from [10], [11], [12], [15], in this paper we give some spectral conditions which is sufficient for a balanced bipartite graphs with large minimum degree to be traceable or hamiltonian. Our results strengthen some results in [10], [11] for n sufficiently large, and whose proofs are provided in Section 3.
Before introducing our results, we need some notations. In order to facilitate understanding, in this paper, when we mention a bipartite graph, we always fix its partite sets, e.g., On,m and Om,n are considered as different bipartite graphs, unless .
Let G1, G2 be two bipartite graphs, with the bipartitions {X1, Y1} and {X2, Y2}, respectively. We use G1⊔G2 to denote the graph obtained from by adding all possible edges between X1 and Y2 and all possible edges between Y1 and X2. We define some classes of graphs as follows: Theorem 1.1 Let G be a balanced bipartite graph of order 2n with δ(G) ≥ k, where k ≥ 0 and . If k ≠ 1 and then G is traceable unless . If and then G is traceable unless .
Theorem 1.2
Let G be a balanced bipartite graph of order 2n with δ(G) ≥ k, where k ≥ 1 and then G is Hamiltonian, unless .
By Perron–Frobenius theorem, .
So, when we immediately have Corollary 1.3 Let G be a balanced bipartite graph on 2n vertices, with minimum degree δ(G) ≥ k, where k ≥ 0 and . If k ≠ 1 and then G is traceable unless . If and then G is traceable unless .
When we immediately have Corollary 1.4 Let G be a balanced bipartite graph of order 2n with δ(G) ≥ k, where k ≥ 1 and then G is Hamiltonian, unless .
Section snippets
Preliminaries
Given a graph G of order n, a vector is called to be defined on G, if there is a 1–1 map φ from V(G) to the entries of ; simply written for each u ∈ V(G). If is an eigenvector of A(G), then is defined on G naturally, xu is the entry of corresponding to the vertex u. One can find thatwhen μ is an eigenvalue of G corresponding to the eigenvector if and only if for each vertex v ∈ V(G). Eq. (2.2) is called the eigenvalue-equation for
Proofs
Proof of Lemma 2.4 Assume that G is a proper subgraph of . By Perron–Frobenius theorem, clearly, we may assume that G is obtained by omitting just one edge uv of . Write X for the set of vertices of of degree k, . Let H be the set of neighbors of all vertices in X, Z for the set of vertices of of degree and Y be the set of the remaining vertices of of degree n. Since δ(G) ≥ k, we see that G must contain all the edges between X and H. Therefore {u, v}⊂Y∪H or {u, v
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Supported by the Natural Science Foundation of China (11871077), the Anhui Natural Science Foundation (1808085MA04), and Natural Science Foundation of Colleges and Universities in Anhui Province (KJ2017A362).