Energy Conditions for Hamiltonicity of Graphs

Abstract

Let be an undirected simple graph of order . Let be the adjacency matrix of , and let be its eigenvalues. The energy of is defined as . Denote by a bipartite graph. In this paper, we establish the sufficient conditions for having a Hamiltonian path or cycle or to be Hamilton-connected in terms of the energy of the complement of , and give the sufficient condition for having a Hamiltonian cycle in terms of the energy of the quasi-complement of .

1. Introduction

We consider only undirected simple graphs. Let be a graph of order with vertex set and edge set . We denote by the number of edges of . The complement of is denoted by , where , , , . For a bipartite graph , the quasi-complement of is denoted by , where , , .

The degree matrix of is denoted by , where denotes the degree of a vertex in the graph . We denote by the maximum degree of . The adjacency matrix of is defined by the matrix of order , where if is adjacent to and otherwise. The signless Laplacian matrix of is defined to be . The Laplacian matrix of is defined to be . In addition, if the graph has no isolated vertices, the normalized Laplacian matrix is defined by . Obviously, , , , and are real symmetric matrix. So their eigenvalues are real numbers and can be ordered. The largest eigenvalue of , denoted by , is said to be the spectral radius of . The largest eigenvalue of , denoted by , is said to be the signless Laplacian spectral radius of . Let be the eigenvalues of . The energy of is defined as .

Let be a graph of order . A Hamiltonian cycle of is a cycle of order contained in . A Hamiltonian path of is a path of order contained in . If contains Hamiltonian cycles, it is said to be Hamiltonian. If every two vertices of are connected by a Hamiltonian path, it is said to be Hamilton-connected. Deciding whether a graph is Hamiltonian is one of the most difficult classical problems in graph theory. Indeed, it is NP-complete.

Lately, the spectral theory of graphs has been wielded to this problem. Fiedler and Nikiforov [1] present some sufficient conditions for a graph having a Hamiltonian path (or cycle) in terms of the spectral radius of the graph or its complements. Zhou [2] studies the signless Laplacian spectral radius of the complements of a graph and gives conditions for the existence of Hamiltonian path or cycle. Li [3] establishes sufficient conditions for a graph having Hamiltonian path or cycle in terms of the energy of the graph. Lu et al. [4] give sufficient conditions for a bipartite graph having Hamiltonian cycles in terms of the spectral radius of the graph. Those results imply that the graphs under discussion should be dense or have two many edges.

For a sparse graph of order , Butler and Chung [5] show that if the nontrivial eigenvalues of the Laplacian matrix of are sufficiently close to the average degree of for sufficiently large , then is Hamiltonian. Fan and Yu [6] get that if the nontrivial eigenvalues of the normalized Laplacian matrix of are sufficiently close to for sufficiently large , then is Hamiltonian.

In this paper, we still study the Hamiltonicity of a dense graph. We provide some sufficient conditions for a graph having a Hamiltonian path or cycle or to be Hamilton-connected in terms of the energy of the complement of the graph (maybe considered as a sparse graph) and present the sufficient conditions for a graph having a Hamiltonian cycle in terms of the energy of the quasi-complement of a bipartite graph. We get the following results, whose proofs are provided in Sections 3 and 4.

Theorem 1. Let be a graph of order . Then(i) contains a Hamiltonian path, if  and ,(ii) contains a Hamiltonian cycle, if  and ,(iii) is Hamilton-connected, if  and .

Theorem 2. Let be a bipartite graph of order , where . Then contains a Hamiltonian cycle, if and .

Li [3] has given some energy conditions for a graph having Hamiltonian paths or cycles as follows.

Theorem 3 (see [3]). Let be a graph of order (≥4). Then(i) contains a Hamiltonian path, if ,(ii) contains a Hamiltonian cycle, if .

Theorem 4 (see [3]). Let be a bipartite graph of order , where . Then contains a Hamiltonian cycle, if

Remark 5. We now compare Theorems 1 and 3, Theorems 2 and 4, respectively.
Firstly, we consider the function . If , So, is monotonously decreasing when . We notice that if , , we have and . Hence, when , , we have So Theorem 1(i) improves Theorem 3(i), when . By a similar discussion, we have that Theorem 1(ii) improves Theorem 3(ii), when .
Secondly, we consider the function . If , So, is monotonously decreasing when . We notice that if and , we have and . Hence, when and , So Theorem 2 improves Theorem 4, when and .

2. Preliminaries

Let be a graph of order . Ore [7] proves that if for any pair of nonadjacent vertices and , then has a Hamiltonian path; if for any pair of nonadjacent vertices and , then has a Hamiltonian cycle. Erdős and Gallai [8] show that if for any pair of nonadjacent vertices and , then is Hamilton-connected.

The idea for the closure of a graph was given by Bondy and Chvátal [9]. For an integer , the -closure of a graph  , denoted by , is the graph obtained from by successively joining pairs of nonadjacent vertices whose degree sum is at least until no such pair remains. The concept of the closure of a balanced bipartite graph is given in [9, 10]. The -closure of a balanced bipartite graph  , where , denoted by , is a graph obtained from by successively joining pairs of nonadjacent vertices and , whose degree sum is at least until no such pairs remain. The -closure of a graph or the -closure of a balanced bipartite graph is unique, independent of the order in which edges are added. We note that for any pair of nonadjacent vertices and of , for any pair of nonadjacent vertices and of .

Lemma 6 (see [9]). (i) A graph of order has a Hamilton path, if and only if has a Hamilton path.
(ii) A graph of order has a Hamilton cycle, if and only if has a Hamilton cycle.
(iii) A graph of order is Hamilton-connected, if and only if is Hamilton-connected.

Lemma 7 (see [10]). A balanced bipartite graph , where , has a Hamiltonian cycle, if and only if has a Hamiltonian cycle.

Lemma 8 (see [11]). Let be any edge in a graph ; one denotes by the subgraph of obtained by deleting the edge . Then .

For a graph , let .

Lemma 9 (see [2]). Let G be a graph with at least one edge. Then

Let be a Hermitian matrix of order and let be the th largest eigenvalue of , .

Lemma 10 (see [12]). Let and be Hermitian matrices of order and let , . If , then

Lemma 11. Let be a graph. Then

Proof. Since , by Lemma 10 Recalling that , , and , the result follows.

Corollary 12. Let be a graph with at least one edge. Then

Proof. The result follows by Lemmas 9 and 11.

3. Proof of Theorem 1

(i) Let . If , then the result follows from (12). Suppose that and does not contain a Hamilton path. Then does not contain a Hamilton path by Lemma 6(i). We notice that for any pair of nonadjacent vertices and (always existing) in . Thus, for any edge , and . By Corollary 12,

By Cauchy-Schwartz inequality, we have that the equality holds if and only if .

Let . If , So, is monotonously decreasing when . We find that when .

Thus

Let . We use to denote the number of -combinations of a set with distinct elements. Then . Because has at least two nonadjacent vertices and such that , then . Hence /.

By Lemma 8, we have that . Thus . Because , we have that a contradiction.

(ii) Let . If , then the result follows from (13). Suppose that and does not contain a Hamilton cycle. Then does not contain a Hamilton cycle too by Lemma 6(ii). By a similar discussion in the proof of Theorem 1(i), we have that for any edge and

By Corollary 12,

We find that when . Using similar arguments as in the proof of Theorem 1(i), we have that

Let . Then . Because has at least two nonadjacent vertices and such that , then . Hence /.

By Lemma 8, we have that . Thus . Because , we have that a contradiction.

(iii) Let . If , then the result follows from (14). Suppose that and is not Hamilton-connected. Then is not Hamilton-connected by Lemma 6(iii). By a similar discussion in the proof of Theorem 1(i), we have that for any edge , and

By Corollary 12,

We notice that when . Using similar arguments as in the proof of Theorem 1(i), we have that

Let . Then . Because has at least two nonadjacent vertices and such that . has at least two nonadjacent vertices and such that , then . Hence .

By Lemma 8, we have that . Thus . Because , we have that a contradiction.

4. Proof of Theorem 2

Proof. Suppose that does not contain a Hamilton cycle. Then does not contain a Hamilton cycle by Lemma 7. Then is not by (13). Observe that for any pair of nonadjacent vertices and (always existing) in . Thus, for any pair of adjacent vertices and in . Then By Corollary 12, we have that
Since is a bipartite graph, . By Cauchy-Schwartz inequality, we have the equality holds if and only if .
Let . If , So, is monotonously decreasing when . We find that when . So Let . Then . Because has two nonadjacent vertices and such that , so . Hence .
By Lemma 8, we have that . Then . Because , we have that a contradiction.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is jointly supported by the National Natural Science Foundation of China under Grant nos. 11071001 and 11071002, the Natural Science Foundation of Anhui Province no. 11040606 M14, and the Natural Science Foundation of Department of Education of Anhui Province no. KJ2011A195.