Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 305164, 6 pages

http://dx.doi.org/10.1155/2014/305164

Research Article

Energy Conditions for Hamiltonicity of Graphs

Guidong Yu,¹ Gaixiang Cai,¹ Miaolin Ye,¹ and Jinde Cao2,3

¹School of Mathematics & Computation Sciences, Anqing Normal College, Anqing 246011, China

²Department of Mathematics, Southeast University, Nanjing 210096, China

³Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 25 December 2013; Accepted 22 January 2014; Published 6 March 2014

Academic Editor: Wenwu Yu

Copyright © 2014 Guidong Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

References

M. Fiedler and V. Nikiforov, “Spectral radius and Hamiltonicity of graphs,” Linear Algebra and Its Applications, vol. 432, no. 9, pp. 2170–2173, 2010.

B. Zhou, “Signless Laplacian spectral radius and Hamiltonicity,” Linear Algebra and Its Applications, vol. 432, no. 2-3, pp. 566–570, 2010.

R. Li, “Energy and some Hamiltonian properties of graphs,” Applied Mathematical Sciences, vol. 3, no. 53–56, pp. 2775–2780, 2009.

M. Lu, H. Q. Liu, and F. Tian, “Spectral radius and Hamiltonian graphs,” Linear Algebra and Its Applications, vol. 437, no. 7, pp. 1670–1674, 2012.

S. Butler and F. Chung, “Small spectral gap in the combinatorial Laplacian implies Hamiltonian,” Annals of Combinatorics, vol. 13, no. 4, pp. 403–412, 2010.

Y. Z. Fan and G. D. Yu, “Spectral condition for a graph to be Hamiltonian with respect to normalized Laplacian,” http://arxiv.org/abs/1207.6824

O. Ore, “Note on Hamilton circuits,” The American Mathematical Monthly, vol. 67, no. 1, p. 55, 1960.

P. Erdős and T. Gallai, “On maximal paths and circuits of graphs,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 10, no. 3-4, pp. 337–356, 1959.

A. Bondy and V. Chvátal, “A method in graph theory,” Discrete Mathematics, vol. 15, no. 2, pp. 111–135, 1976.

G. Hendry, “Extending cycles in bipartite graphs,” Journal of Combinatorial Theory B, vol. 51, no. 2, pp. 292–313, 1991.

J. Day and W. So, “Singular value inequality and graph energy change,” Electronic Journal of Linear Algebra, vol. 16, pp. 291–299, 2007.

W. So, “Commutativity and spectra of Hermitian matrices,” Linear Algebra and Its Applications, vol. 212-213, no. 15, pp. 121–129, 1994.