Let G be an undirected simple graph of order n. Let A(G) be the adjacency matrix of G, and let μ1(G)≤μ2(G)≤⋯≤μn(G) be its eigenvalues. The energy of G is defined as ℰ(G)=∑i=1n|μi(G)|. Denote by GBPT a bipartite graph. In this paper, we establish the sufficient conditions for G having a Hamiltonian path or cycle or to be Hamilton-connected in terms of the energy of the complement of G, and give the sufficient condition for GBPT having a Hamiltonian cycle in terms of the energy of the quasi-complement of GBPT.

1. Introduction

We consider only undirected simple graphs. Let G=(V,E) be a graph of order n with vertex set V=V(G)={v1,v2,…,vn} and edge set E=E(G). We denote by e(G) the number of edges of G. The complement of G is denoted by Gc:=(V,Ec), where Ec:={xy:x∈V, y∈V, x≠y, xy∉E}. For a bipartite graph GBPT=(X,Y;E), the quasi-complement of GBPT is denoted by GBPT*:=(X,Y;E′), where E′={xy:x∈X, y∈Y, xy∉E}.

The degree matrix of G is denoted by D(G)=diag(dG(v1),dG(v2),…,dG(vn)), where dG(v) denotes the degree of a vertex v in the graph G. We denote by Δ(G) the maximum degree of G. The adjacency matrix of G is defined by the matrix A(G)=[aij] of order n, where aij=1 if vi is adjacent to vj and aij=0 otherwise. The signless Laplacian matrix of G is defined to be Q(G)=D(G)+A(G). The Laplacian matrix of G is defined to be L(G)=D(G)-A(G). In addition, if the graph G has no isolated vertices, the normalized Laplacian matrix is defined by ℒ(G)=D(G)-1/2L(G)D(G)-1/2. Obviously, A(G), Q(G), L(G), and ℒ(G) are real symmetric matrix. So their eigenvalues are real numbers and can be ordered. The largest eigenvalue of A(G), denoted by μ(G), is said to be the spectral radius of G. The largest eigenvalue of Q(G), denoted by q(G), is said to be the signless Laplacian spectral radius of G. Let μ1(G)≤μ2(G)≤⋯≤μn(G) be the eigenvalues of A(G). The energy of G is defined as ℰ(G)=∑i=1n|μi(G)|.

Let G be a graph of order n. A Hamiltonian cycle of G is a cycle of order n contained in G. A Hamiltonian path of G is a path of order n contained in G. If G contains Hamiltonian cycles, it is said to be Hamiltonian. If every two vertices of G 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 G of order n, Butler and Chung [5] show that if the nontrivial eigenvalues of the Laplacian matrix of G are sufficiently close to the average degree of G for sufficiently large n, then G is Hamiltonian. Fan and Yu [6] get that if the nontrivial eigenvalues of the normalized Laplacian matrix of G are sufficiently close to 1 for sufficiently large n, then G 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 G be a graph of order n≥3. Then

G contains a Hamiltonian path, if
(1)ℰ(Gc)>2e(Gc)-n+2-Δ(Gc)+(n-1)(2e(Gc)-(n-Δ(Gc))2),

and n-2e(Gc)≤Δ(Gc)≤n-n-1,

G contains a Hamiltonian cycle, if
(2)ℰ(Gc)>2e(Gc)-n+3-Δ(Gc)+(n-1)(2e(Gc)-(n-1-Δ(Gc))2),

and n-1-2e(Gc)≤Δ(Gc)≤n-1-n-1,

G is Hamilton-connected, if
(3)ℰ(Gc)>2e(Gc)-n+4-Δ(Gc)+(n-1)(2e(Gc)-(n-2-Δ(Gc))2),

and n-2-2e(Gc)≤Δ(Gc)≤n-2-n-1.

Theorem 2.

Let GBPT=(X,Y;E) be a bipartite graph of order n=2r≥4, where |X|=|Y|=r≥2. Then GBPT contains a Hamiltonian cycle, if
(4)ℰ(GBPT*)>2e(GBPT*)+2-2Δ(GBPT*)+(n-2)(2e(GBPT*)-2(r-Δ(GBPT*))2),
and r-e(GBPT*)≤Δ(GBPT*)≤r-r.

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

Theorem 3 (see [<xref ref-type="bibr" rid="B9">3</xref>]).

Let G be a graph of order n(≥4). Then

G contains a Hamiltonian path, if ℰ(Gc)>2e(Gc)-2n+2+e(Gc)(n-1+1),

G contains a Hamiltonian cycle, if ℰ(Gc)>2e(Gc)-2n+4+(n-1)e(Gc)/n(n+1+1).

Theorem 4 (see [<xref ref-type="bibr" rid="B9">3</xref>]).

Let GBPT=(X,Y;E) be a bipartite graph of order n=2r≥4, where |X|=|Y|=r≥2. Then GBPT contains a Hamiltonian cycle, if
(5)ℰ(GBPT*)>2e(GBPT*)-2r+2+e(GBPT*)(n-2+2).

Remark 5.

We now compare Theorems 1 and 3, Theorems 2 and 4, respectively.

Firstly, we consider the function f(x)=x+(n-1)(2e(Gc)-x2). If 2e(Gc)/n<x<2e(Gc),
(6)f′(x)=1+-(n-1)x(n-1)(2e(Gc)-x2)<0.
So, f(x) is monotonously decreasing when 2e(Gc)/n<x≤2e(Gc). We notice that if n-2e(Gc)≤Δ(Gc)≤n/2, n≥3, we have
(7)2e(Gc)n<e(Gc)≤nΔ(Gc)2≤n-Δ(Gc)≤2e(Gc),
and Δ(Gc)≤n-n-1. Hence, when n-2e(Gc)≤Δ(Gc)≤n/2, n≥3, we have
(8)2e(Gc)-2n+2+e(Gc)(n-1+1)=2e(Gc)-2n+2+f(e(Gc))≥2e(Gc)-2n+2+f(n-Δ(Gc))=2e(Gc)-n+2-Δ(Gc)+(n-1)(2e(Gc)-(n-Δ(Gc))2).
So Theorem 1(i) improves Theorem 3(i), when n-2e(Gc)≤Δ(Gc)≤n/2. By a similar discussion, we have that Theorem 1(ii) improves Theorem 3(ii), when n-1-2e(Gc)≤Δ(Gc)≤(n-1)/2.

Secondly, we consider the function g(x)=2x+(n-2)(2e(GBPT*)-2x2). If 2e(GBPT*)/n<x<e(GBPT*),
(9)g′(x)=2+-2(n-2)x(n-2)(2e(GBPT*)-2x2)<0.
So, g(x) is monotonously decreasing when 2e(GBPT*)/n<x≤e(GBPT*). We notice that if r-e(GBPT*)≤Δ(GBPT*)≤r/2 and r≥2, we have
(10)2e(GBPT*)n<e(GBPT*)2≤2rΔ(GBPT*)4≤r-Δ(GBPT*)≤e(GBPT*)
and Δ(GBPT*)≤r-r. Hence, when r-e(GBPT*)≤Δ(GBPT*)≤r/2 and r≥2,
(11)2e(GBPT*)-2r+2+e(GBPT*)(n-2+2)=2e(GBPT*)-2r+2+g(e(GBPT*)2)≥2e(GBPT*)-2r+2+g(r-Δ(GBPT*))=2e(GBPT*)+2-2Δ(GBPT*)+(n-2)(2e(GBPT*)-2(r-Δ(GBPT*))2).
So Theorem 2 improves Theorem 4, when r-e(GBPT*)≤Δ(GBPT*)≤r/2 and r≥2.

2. Preliminaries

Let G be a graph of order n. Ore [7] proves that if
(12)dG(u)+dG(v)≥n-1
for any pair of nonadjacent vertices u and v, then G has a Hamiltonian path; if
(13)dG(u)+dG(v)≥n
for any pair of nonadjacent vertices u and v, then G has a Hamiltonian cycle. Erdős and Gallai [8] show that if
(14)dG(u)+dG(v)≥n+1
for any pair of nonadjacent vertices u and v, then G is Hamilton-connected.

The idea for the closure of a graph G was given by Bondy and Chvátal [9]. For an integer k≥0, the k-closure of a graphG, denoted by 𝒞k(G), is the graph obtained from G by successively joining pairs of nonadjacent vertices whose degree sum is at least k until no such pair remains. The concept of the closure of a balanced bipartite graph is given in [9, 10]. The k-closure of a balanced bipartite graphGBPT:=(X,Y;E), where |X|=|Y|, denoted by 𝒞k(GBPT), is a graph obtained from GBPT by successively joining pairs of nonadjacent vertices x∈X and y∈Y, whose degree sum is at least k until no such pairs remain. The k-closure of a graph G or the k-closure of a balanced bipartite graph GBPT is unique, independent of the order in which edges are added. We note that d𝒞k(G)(u)+d𝒞k(G)(v)≤k-1 for any pair of nonadjacent vertices u and v of 𝒞k(G), d𝒞k(GBPT)(x)+d𝒞k(GBPT)(y)≤k-1 for any pair of nonadjacent vertices x∈X and y∈Y of 𝒞k(GBPT).

Lemma 6 (see [<xref ref-type="bibr" rid="B1">9</xref>]).

(i) A graph G of order n has a Hamilton path, if and only if 𝒞n-1(G) has a Hamilton path.

(ii) A graph G of order n has a Hamilton cycle, if and only if 𝒞n(G) has a Hamilton cycle.

(iii) A graph G of order n is Hamilton-connected, if and only if 𝒞n+1(G) is Hamilton-connected.

Lemma 7 (see [<xref ref-type="bibr" rid="B8">10</xref>]).

A balanced bipartite graph GBPT=(X,Y;E), where |X|=|Y|=r≥2, has a Hamiltonian cycle, if and only if 𝒞r+1(GBPT) has a Hamiltonian cycle.

Lemma 8 (see [<xref ref-type="bibr" rid="B4">11</xref>]).

Let e be any edge in a graph G; one denotes by G-e the subgraph of G obtained by deleting the edge e. Then ℰ(G)-2≤ℰ(G-e)≤ℰ(G)+2.

For a graph G, let Z(G):=∑uv∈E(G)(dG(u)+dG(v))=∑u∈V(G)dG2(u).

Lemma 9 (see [<xref ref-type="bibr" rid="B13">2</xref>]).

Let G be a graph with at least one edge. Then
(15)q(G)≥Z(G)e(G).

Let M be a Hermitian matrix of order n and let λi(M) be the ith largest eigenvalue of M, 1≤i≤n.

Lemma 10 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let B and C be Hermitian matrices of order n and let 1≤i, j≤n. If i+j≤n+1, then
(16)λi(B)+λj(C)≥λi+j-1(B+C).

Lemma 11.

Let G be a graph. Then
(17)μ(G)≥q(G)-Δ(G).

Proof.

Since Q(G)=A(G)+D(G), by Lemma 10(18)λ1(A(G))+λ1(D(G))≥λ1(Q(G)).
Recalling that λ1(A(G))=μ(G), λ1(D(G))=Δ(G), and q(G)=λ1(Q(G)), the result follows.

Corollary 12.

Let G be a graph with at least one edge. Then
(19)μ(G)≥Z(G)e(G)-Δ(G).

Proof.

The result follows by Lemmas 9 and 11.

3. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>

(i) Let H:=𝒞n-1(G). If H=Kn, then the result follows from (12). Suppose that H≠Kn and G does not contain a Hamilton path. Then H does not contain a Hamilton path by Lemma 6(i). We notice that dH(u)+dH(v)≤n-2 for any pair of nonadjacent vertices u and v (always existing) in H. Thus, for any edge uv∈E(Hc),
(20)dHc(u)+dHc(v)=2(n-1)-[dH(u)+dH(v)]≥n,
and Z(Hc)=∑uv∈E(Hc)[dHc(u)+dHc(v)]≥ne(Hc). By Corollary 12,
(21)μ(Hc)≥Z(Hc)e(Hc)-Δ(Hc)≥n-Δ(Hc)≥n-Δ(Gc).

By Cauchy-Schwartz inequality, we have that
(22)ℰ(Hc)=∑i=1n|μi(Hc)|=μ(Hc)+∑i=1n-1|μi(Hc)|≤μ(Hc)+(n-1)∑i=1n-1μi2(Hc)=μ(Hc)+(n-1)(∑i=1nμi2(Hc)-μ2(Hc))=μ(Hc)+(n-1)(2e(Hc)-μ2(Hc));
the equality holds if and only if |μ1(Hc)|=|μ2(Hc)|=⋯=|μn-1(Hc)|.

Let f(x)=x+(n-1)(2e(Hc)-x2). If 2e(Hc)/n<x<2e(Hc),
(23)f′(x)=1+-(n-1)x(n-1)(2e(Hc)-x2)<0.
So, f(x) is monotonously decreasing when 2e(Hc)/n<x<2e(Hc). We find that
(24)2e(Hc)n≤2e(Gc)n≤Δ(Gc)<n-Δ(Gc)≤μ(Hc)<2e(Hc),
when Δ(Gc)≤n-n-1.

Let s:=e(H)-e(G). We use C(n,k) to denote the number of k-combinations of a set with n distinct elements. Then e(Gc)-e(Hc)=(C(n,2)-e(G))-(C(n,2)-e(H))=s. Because H has at least two nonadjacent vertices u and v such that dH(u)+dH(v)≤n-2, then e(H)≤(n-2)+C(n-2,2)=(n2-3n+2)/2. Hence s≤(n2-3n+2)/2-e(G)=(n2-3n+2)/2-(C(n,2)-e(Gc))=e(Gc)-n+1.

By Lemma 8, we have that ℰ(Hc)≥ℰ(Gc)-2s. Thus ℰ(Hc)≥ℰ(Gc)-2e(Gc)+2n-2. Because e(Hc)≤e(Gc), we have that
(26)ℰ(Gc)≤2e(Gc)-n+2-Δ(Gc)+(n-1)(2e(Gc)-(n-Δ(Gc))2),
a contradiction.

(ii) Let H-=𝒞n(G). If H-=Kn, then the result follows from (13). Suppose that H-≠Kn and G does not contain a Hamilton cycle. Then H- 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 dH-c(u)+dH-c(v)≥n-1 for any edge uv∈E(H-c) and
(27)Z(H-c)=∑uv∈E(H-c)[dH-c(u)+dH-c(v)]≥(n-1)e(H-c).

By Corollary 12,
(28)μ(H-c)≥Z(H-c)e(H-c)-Δ(H-c)≥n-1-Δ(H-c)≥n-1-Δ(Gc).

We find that
(29)2e(H-c)n≤2e(Gc)n≤Δ(Gc)<n-1-Δ(Gc)≤μ(H-c)<2e(H-c),
when Δ(Gc)≤n-1-n-1. Using similar arguments as in the proof of Theorem 1(i), we have that
(30)ℰ(H-c)≤f(μ(H-c))≤f(n-1-Δ(Gc))=n-1-Δ(Gc)+(n-1)(2e(H-c)-(n-1-Δ(Gc))2).

Let s:=e(H-)-e(G). Then e(Gc)-e(H-c)=(C(n,2)-e(G))-(C(n,2)-e(H-))=s. Because H- has at least two nonadjacent vertices u and v such that dH-(u)+dH-(v)≤n-1, then e(H-)≤(n-1)+C(n-2,2)=(n2-3n+4)/2. Hence s≤(n2-3n+4)/2-e(G)=(n2-3n+4)/2-(C(n,2)-e(Gc))=e(Gc)-n+2.

By Lemma 8, we have that ℰ(H-c)≥ℰ(Gc)-2s. Thus ℰ(H-c)≥ℰ(Gc)-2e(Gc)+2n-4. Because e(H-c)≤e(Gc), we have that
(31)ℰ(Gc)≤2e(Gc)-n+3-Δ(Gc)+(n-1)(2e(Gc)-(n-1-Δ(Gc))2),
a contradiction.

(iii) Let H^=𝒞n+1(G). If H^=Kn, then the result follows from (14). Suppose that H^≠Kn and G is not Hamilton-connected. Then H^ is not Hamilton-connected by Lemma 6(iii). By a similar discussion in the proof of Theorem 1(i), we have that dH^c(u)+dH^c(v)≥n-2 for any edge uv∈E(H^c), and
(32)Z(H^c)=∑uv∈E(H^c)[dH^c(u)+dH^c(v)]≥(n-2)e(H^c).

By Corollary 12,
(33)μ(H^c)≥Z(H^c)e(H^c)-Δ(H^c)≥n-2-Δ(H^c)≥n-2-Δ(Gc).

We notice that
(34)2e(H^c)n≤2e(Gc)n≤Δ(Gc)<n-2-Δ(Gc)≤μ(H^c)<2e(H^c),
when Δ(Gc)≤n-2-n-1. Using similar arguments as in the proof of Theorem 1(i), we have that
(35)ℰ(H^c)≤f(μ(H^c))≤f(n-2-Δ(Gc))=n-2-Δ(Gc)+(n-1)(2e(H^c)-(n-2-Δ(Gc))2).

Let s:=e(H^)-e(G). Then e(Gc)-e(H^c)=(C(n,2)-e(G))-(C(n,2)-e(H^))=s. Because H^ has at least two nonadjacent vertices u and v such that dH^(u)+dH^(v)≤n.H^ has at least two nonadjacent vertices u and v such that dH^(u)+dH^(v)≤n, then e(H^)≤n+C(n-2,2)=(n2-3n+6)/2. Hence s≤(n2-3n+6)/2-e(G)=(n2-3n+6)/2-(C(n,2)-e(Gc))=e(Gc)-n+3.

By Lemma 8, we have that ℰ(H^c)≥ℰ(Gc)-2s. Thus ℰ(H^c)≥ℰ(Gc)-2e(Gc)+2n-6. Because e(H^c)≤e(Gc), we have that
(36)ℰ(Gc)≤2e(Gc)-n+4-Δ(Gc)+(n-1)(2e(Gc)-(n-2-Δ(Gc))2),
a contradiction.

4. Proof of Theorem <xref ref-type="statement" rid="thm1.2">2</xref> Proof.

Suppose that GBPT does not contain a Hamilton cycle. Then HBPT:=𝒞r+1(GBPT) does not contain a Hamilton cycle by Lemma 7. Then HBPT is not Kr,r by (13). Observe that dHBPT(x)+dHBPT(y)≤r for any pair of nonadjacent vertices x∈X and y∈Y (always existing) in HBPT. Thus,
(37)dHBPT*(x)+dHBPT*(y)=r-dHBPT(x)+r-dHBPT(y)≥r,
for any pair of adjacent vertices x∈X and y∈Y in HBPT*. Then
(38)Z(HBPT*)=∑u∈V(HBPT*)dHBPT*2(u)=∑xy∈E(HBPT*)(dHBPT*(x)+dHBPT*(y))≥re(HBPT*).
By Corollary 12, we have that
(39)μ(HBPT*)≥Z(HBPT*)e(HBPT*)-Δ(HBPT*)≥r-Δ(HBPT*)≥r-Δ(GBPT*).

Since HBPT* is a bipartite graph, μn(HBPT*)=-μ1(HBPT*). By Cauchy-Schwartz inequality, we have
(40)ℰ(HBPT*)=∑i=1n|μi(HBPT*)|=2μ(HBPT*)+∑i=2n-1|μi(HBPT*)|≤2μ(HBPT*)+(n-2)∑i=2n-1μi2(HBPT*)=2μ(HBPT*)+(n-2)(∑i=1nμi2(HBPT*)-2μ2(HBPT*))=2μ(HBPT*)+(n-2)(2e(HBPT*)-2μ2(HBPT*));
the equality holds if and only if |μ2(HBPT*)|=⋯=|μn-1(HBPT*)|.

Let g(x)=2x+(n-2)(2e(HBPT*)-2x2). If 2e(HBPT*)/n<x<e(HBPT*),
(41)g′(x)=2+-2(n-2)x(n-2)(2e(HBPT*)-2x2)<0.
So, g(x) is monotonously decreasing when 2e(HBPT*)/n<x<e(HBPT*). We find that
(42)2e(HBPT*)n≤Δ(GBPT*)<r-Δ(GBPT*)≤μ(HBPT*)<e(HBPT*),
when Δ(Gc)≤r-r. So
(43)ℰ(HBPT*)≤g(μ(HBPT*))≤g(r-Δ(GBPT*))=2(r-Δ(GBPT*))+(n-2)(2e(HBPT*)-2(r-Δ(GBPT*))2).
Let t:=e(HBPT)-e(GBPT). Then e(GBPT*)-e(HBPT*)=(r2-e(GBPT))-(r2-e(HBPT))=t. Because HBPT has two nonadjacent vertices x∈X and y∈Y such that dHBPT(x)+dHBPT(y)≤r, so e(HBPT)≤(r-1)2+r=r2-r+1. Hence t≤r2-r+1-e(GBPT)=r2-r+1-(r2-e(GBPT*))=e(GBPT*)-r+1.

By Lemma 8, we have that ℰ(HBPT*)≥ℰ(GBPT*)-2t. Then ℰ(HBPT*)≥ℰ(GBPT*)-2e(GBPT*)+2r-2. Because e(HBPT*)≤e(GBPT*), we have that
(44)ℰ(GBPT*)≤2e(GBPT*)+2-2Δ(GBPT*)+(n-2)(2e(GBPT*)-2(r-Δ(GBPT*))2),
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.

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