JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 10.1155/2014/282053 282053 Research Article Signless Laplacian Spectral Conditions for Hamiltonicity of Graphs Yu Guidong 1 Ye Miaolin 1 Cai Gaixiang 1 http://orcid.org/0000-0003-3133-7119 Cao Jinde 2, 3 Soldovieri Francesco 1 School of Mathematics & Computation Sciences Anqing Normal College, Anqing 246011 China 2 Department of Mathematics Southeast University, Nanjing, Jiangsu 210096 China seu.edu.cn 3 Department of Mathematics, Faculty of Science King Abdulaziz University, Jeddah 21589 Saudi Arabia kau.edu.sa 2014 3062014 2014 23 01 2014 07 06 2014 16 06 2014 30 6 2014 2014 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.

We establish some signless Laplacian spectral radius conditions for a graph to be Hamiltonian or traceable or Hamilton-connected.

1. Introduction

Let a graph, G=(V,E) be a simple graph of order n with vertex set V={v1,v2,,vn} and edge set E. Denote by e(G)=|E| the number of edges of the graph G. Write by Kn a complete graph of order n, On an empty graph of order n (without edges), and Kn,m a complete bipartite graph with two parts having n, m vertices, respectively. The graph G is said to be Hamiltonian, if it has a Hamiltonian cycle which is a cycle of order n contained in G. The graph G is said to be traceable if it has a Hamiltonian path which is a path of order n contained in G. The problem of deciding whether a graph is Hamiltonian is Hamiltonian problem, which is one of the most difficult classical problems in graph theory. Indeed, it is NP-complete problem.

The adjacency matrix of G is defined to be a matrix A(G)=[aij]n×n, where aij=1 if vi is adjacent to vj and aij=0 otherwise. The largest eigenvalue of A(G) is called to be the spectral radius of G, which is denoted by μ(G). The degree matrix of G is written by D(G)=diag(dG(v1),dG(v2),,dG(vn)), where dG(vi)(i=1,2,,n) denotes the degree of the vertex vi in the graph G. The signless Laplacian matrix of G is defined by Q(G)=D(G)+A(G). The largest eigenvalue of Q(G) is called to be the signless Laplacian spectral radius of G, which is denoted by q(G).

Recently, using spectral graph theory to study the Hamiltonian problem has received a lot of attention. Some spectral conditions for a graph to be Hamiltonian or traceable have been given in . In this paper, we still study the Hamiltonicity of a graph. Firstly, we present a signless Laplacian spectral radius condition for a bipartite graph to be Hamiltonian in Section 2. Secondly, we give some signless Laplacian spectral radius conditions for a graph to be traceable or Hamilton-connected in Section 3 and Section 4, respectively.

2. Signless Laplacian Spectral Radius in Hamiltonian Bipartite Graphs

The definition of the closure of a balanced bipartite graph can be found in [7, 8]. For a positive integer k, the k-closure of a balanced bipartite graph GBPT=(X,Y;E), where |X|=|Y|, written by Ck(GBPT), is a graph obtained from GBPT by successively joining pairs of nonadjacent vertices xX and yY, whose degree sum is at least k, until no such pairs remain. By the definition of the Ck(GBPT), we have that dCk(GBPT)(x)+dCk(GBPT)(y)k-1 for any pair of nonadjacent vertices xX and yY of Ck(GBPT).

Lemma 1 (see [<xref ref-type="bibr" rid="B4">9</xref>]).

Let GBPT=(X,Y;E) be a connected balanced bipartite graph, where |X|=|Y|=r2. Then, GBPT is Hamiltonian if and only if Cr+1(GBPT) is Hamiltonian.

For a graph G, write Z(G)=uvE(G)(dG(u)+dG(v))=uV(G)dG2(u), and let Δ(G) be maximum degree of G. A regular graph is a graph for which every vertex in the graph has the same degree. A semi-regular graph is a bipartite graph for which every vertex in the same partite set has the same degree.

Lemma 2 (see [<xref ref-type="bibr" rid="B14">2</xref>]).

Let G be a graph with at least one edge. Then, (1)q(G)Z(G)e(G), if and only if G is regular or semi-regular.

Let M be a Hermitian matrix of order n, and let λ1(M)λ2(M)λn(M) be the eigenvalues of M.

Lemma 3 (see [<xref ref-type="bibr" rid="B13">10</xref>]).

Let B and C be Hermitian matrices of order n, 1i, jn. Then, (2)λi(B)+λj(C)λi+j-1(B+C), if i+jn+1.

Lemma 4.

Let G be a graph. Then, (3)q(G)μ(G)+Δ(G).

Proof.

Because Q(G)=A(G)+D(G), by Lemma 3, (4)λ1(A(G))+λ1(D(G))λ1(Q(G)). We notice that λ1(A(G))=μ(G), λ1(D(G))=Δ(G), and λ1(Q(G))=q(G). So, the result follows.

Let GBPT=(X,Y;E) be a bipartite graph, the quasi-complement of GBPT is denoted by GBPT*=(X,Y;E), where E={xy:xX,yY,xyE}.

Theorem 5.

Let GBPT=(X,Y;E) be a connected balanced bipartite graph, where |X|=|Y|=r2. If (5)q(GBPT*)<r, then GBPT is Hamiltonian.

Proof.

Suppose that GBPT is not Hamiltonian. Then, HBPT=Cr+1(GBPT) is not Hamiltonian too by Lemma 1, and therefore, HBPT is not Kr,r. Thus, there exists a vertex xX and a vertex yY such that xyE(HBPT). We find that dHBPT(x)+dHBPT(y)r for any pair of nonadjacent vertices xX and yY in HBPT. So, (6)dHBPT*(x)+dHBPT*(y)=r-dHBPT(x)+r-dHBPT(y)r, for any pair of adjacent vertices xX, yY in HBPT*. Hence, (7)Z(HBPT*)=xyE(HBPT*)(dHBPT*(x)+dHBPT*(y))re(HBPT*).

By Lemma 2, we have that (8)q(HBPT*)Z(HBPT*)e(HBPT*)r.

As HBPT* is a subgraph of GBPT*, by Perron-Frobenius theorem, (9)q(GBPT*)q(HBPT*).

Thus, by (5), (8), and (9), we have that (10)r>q(GBPT*)q(HBPT*)Z(HBPT*)e(HBPT*)r, a contradiction.

Li  has given a sufficient condition for a bipartite graph to be Hamiltonian as follows.

Theorem 6 (see [<xref ref-type="bibr" rid="B12">4</xref>]).

Let GBPT=(X,Y;E) be a connected balanced bipartite graph, where |X|=|Y|=r2. If (11)μ(GBPT*)r-22, then GBPT is Hamiltonian.

Remark 7.

We now compare Theorems 5 and 6. If μ(GBPT*)(r-2)/2 and Δ(GBPT*)<r-(r-2)/2, we have that q(GBPT*)<r by Lemma 4. Hence Theorem 5 improves Theorem 6 when Δ(GBPT*)<r-(r-2)/2. For example, let GBPT be a regular connected balanced bipartite graph with degree (r+1)/2, where r is odd and |X|=|Y|=r6. Then, its quasi-complement GBPT* is a regular graph with degrees (r-1)/2, μ(GBPT*)=(r-1)/2, and q(GBPT*)=r-1. GBPT satisfies the condition of Theorems 5, and hence, it is Hamiltonian. But it does not satisfy the condition of Theorem 6.

3. Signless Laplacian Spectral Radius in Traceable Graphs

Write Kn-1+v for Kn-1 together with an isolated vertex. Let G=(V(G),E(G)) and H=(V(H),E(H)) be two disjoint graphs. The disjoint union of G and H, denoted by GH, is the graph with vertex set V(G)V(H) and edge set E(G)E(H). If G1Gk, we write kG1 for G1Gk. The join of G and H, denoted by GH, is the graph obtained from GH by adding edges joining every vertex of G to every vertex of H.

Lemma 8 (see [<xref ref-type="bibr" rid="B11">3</xref>]).

Let G be a connected graph of order n4. If (12)e(G)(n-2)(n-3)2+2, then G is traceable unless GK1(Kn-32K1), K2(3K1K2), or K4(6K1).

Let G be a graph containing a vertex v. Denote mG(v)=m(v)=(1/dG(v))uNG(v)dG(u) if dG(v)>0 and mG(v)=0 otherwise, where NG(v) or simply N(v) denotes the neighborhood of v in G.

Lemma 9 (see [<xref ref-type="bibr" rid="B5">11</xref>]).

Let G be a graph of order n. Then, (13)max{dG(v)+mG(v):vV(G)}2e(G)n-1+n-2, with equality if and only if GK1,n-1 or G=Kn-1+v.

Lemma 10 (see [<xref ref-type="bibr" rid="B3">12</xref>]).

Let G be a connected graph. Then (14)q(G)max{dG(v)+mG(v):vV(G)}, with equality if and only if G is a regular graph or a semi-regular graph.

In fact, if G is disconnected, there exists a component H of G such that (15)q(G)=q(H)max{dH(v)+mH(v):vV(H)}max{dG(v)+mG(v):vV(G)}. So the inequality (14) also holds when G is a disconnected graph. By Lemmas 9 and 10, we have the following result; also see .

Corollary 11.

Let G be a graph of order n. Then, (16)q(G)2e(G)n-1+n-2. If G is connected, then the equality in (16) holds if and only if G=K1,n-1 or G=Kn. Otherwise, the equality in (16) holds if and only if G=Kn-1+v.

Given a graph G of order n, a vector XRn is called a function defined on G, if there is a 1-1 map φ from V(G) to the entries of X, simply written as Xu=φ(u) for each uV(G), Xu is also called the value of u given by X. If X is an eigenvector of Q(G) corresponding to the eigenvalue q, then X is defined naturally on G; that is, Xu is the entry of X corresponding to the vertex u. One can find that (17)[q-dG(v)]Xv=uNG(v)Xu,foreachvV(G), where NG(v) denotes the neighborhood of v in G. The equation (17) is called (q,X)-eigenequation of G.

Theorem 12.

Let G be a connected graph of order n4. If (18)q(G)2(n-2)2+4n-1, then G is traceable.

Proof.

By Corollary 11 and (18), we have (19)e(G)(n-1)q(G)-(n-1)(n-2)2(n-2)(n-3)2+2.

Suppose that G is non-traceable. Then, by Lemma 8 and (19), GK1(Kn-32K1), K2(3K1K2), or K4(6K1).

If GK1(Kn-32K1), let X=(X1,X2,,Xn)T be the eigenvector of Q(G) corresponding to eigenvalue q(G). By (18), we know that q(G)1, n-4. Thus, by (17), all vertices of degree 1 have the same values given by X, say X1; all vertices of degree n-3 have the same values by X, say X2. Denote by X3 the value of the vertex of degree n-1 given by X. Also, by (17), we have (20)(q(G)-1)X1=X3,(q(G)-(n-3))X2=(n-4)X2+X3,(q(G)-(n-1))X3=2X1+(n-3)X2. Transform (20) into a matrix equation (B-q(G)I)X=0, where X=(X1,X2,X3)T and (21)B=[10102n-712n-3n-1].

Thus, q(G) is the largest root of the following equation: (22)q3+(-3n+7)q2+(2n2-7n)q-2n2+14n-24=0. Let f(x)=x3+(-3n+7)x2+(2n2-7n)x-2n2+14n-24; then f(x)=3x2+2(-3n+7)x+2n2-7n. Let f(x)=0; we have two values x1 and x2, such that f(x1)=f(x2)=0, where (23)x1=3n-7-3n2-21n+493,x2=3n-7+3n2-21n+493.

Hence, f(x) is strictly increasing with respect to x for x>x2.

Because f(2(n-3))=2n2-17n+33>0 and (2(n-2)2+4)/(n-1)>2(n-3)>x2, we have that f((2(n-2)2+4)/(n-1))>0, which implies that q(G)<(2(n-2)2+4)/(n-1).

If GK2(3K1K2), let X=(X1,X2,,X7)T be the eigenvector of Q(G) corresponding to eigenvalue q(G). By (18), we know that q(G)2,5. Thus, by (17), three vertices of degree 2 have the same values given by X, say X1; two vertices of degree 3 have the same values, say X2; two vertices of degree 6 have the same values, say X3. Also, by (17), we have (24)(q(G)-2)X1=2X3,(q(G)-3)X2=X2+2X3,(q(G)-6)X3=3X1+2X2+X3. Transform (24) into a matrix equation (B-q(G)I)X=0, where X=(X1,X2,X3)T and (25)B=.

Thus, q(G) is the largest root of the following equation: (26)q3-13q2+40q-24=0. Let g(x)=x3-13x2+40x-24; we can easily get that g(x) is strictly increasing with respect to x for x>20/3.

Consider g(9)=12>0, which implies that q(G)<9.

If GK4(6K1), we easily calculate q(G)=8+210<44/3.

Thus, in either case, we have a contradiction.

Lu et al.  have given a sufficient condition for a graph to be traceable as follows.

Theorem 13 (see [<xref ref-type="bibr" rid="B11">3</xref>]).

Let G be a connected graph of order n5. If (27)μ(G)(n-3)2+2, then G is traceable.

Example 14.

There are graphs to which Theorem 12 may apply but Theorem 13 may not. Let G=(KrKr)K1 of order n=2r+1, where r4. Surely, the graph G is traceable. By a little computation, μ(G) is the largest root of the polynomial f(x)=x[x-(r-1)]-2r and q(G) is the largest root of the polynomial g(x)=[x-(2r-1)](x-2r)-2r. Hence, (28)μ(G)=r+1+r2+6r+12<4r2-8r+6=(n-3)2+2,q(G)=4r-14>(2r-1)2+2r=2(n-2)2+4n-1. So, we can apply Theorem 12 but not Theorem 13 for G to be traceable.

4. Signless Laplacian Spectral Radius in Hamilton-Connected Graphs

For a graph G of order n, Erdös and Gallai  prove that if (29)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 can be found in . For a positive integer k, the k-closure of a graph G=(V,E), denoted by Ck(G), is a graph obtained from G by successively joining pairs of nonadjacent vertices uV and vV, whose degree sum is at least k until no such pairs remain. By the definition of the k-closure of G, we have that dCk(G)(u)+dCk(G)(v)k-1 for any pair of nonadjacent vertices uV and vV of Ck(G).

Lemma 15 (see [<xref ref-type="bibr" rid="B1">7</xref>]).

Let G be a graph of order n. Then, G is Hamilton-connected if and only if Cn+1(G) is Hamilton-connected.

Lemma 16.

Let G be a simple graph with degree sequence (dG(v1),dG(v2),,dG(vn)), where dG(v1)dG(v2)dG(vn) and n3. Suppose that there is no integer kn/2 such that dG(vk-1)k and dG(vn-k)n-k. Then, G is Hamilton-connected.

Proof.

Let H-=Cn+1(G) be the (n+1)-closure of G. Next, we will prove that H- is a complete graph; then the result follows according to (29). To the contrary, suppose that H- is not a complete graph, and let u and v be two nonadjacent vertices in H- with (30)dH-(u)dH-(v) and dH-(u)+dH-(v) being as large as possible. By the definition of Cn+1(G), we have (31)dH-(u)+dH-(v)n. Denote by S the set of vertices in V{v} which are nonadjacent to v in H-. Denote by T the set of vertices in V{u} which are nonadjacent to u in H-. Then, (32)|S|=n-1-dH-(v),|T|=n-1-dH-(u). Furthermore, by dH-(u)+dH-(v) being as large as possible, each vertex in S has degree at most dH-(u) and each vertex in T{u} has degree at most dH-(v). Let k=dH-(u). According to (31) and (32), we have that |S|=n-1-dH-(v)dH-(u)-1=k-1, |T|+1=n-1-dH-(u)+1=n-dH-(u)=n-k. Then H- has at least k-1 vertices of degree not exceeding k and at least n-k vertices of degree not exceeding n-k. Because G is a spanning subgraph of H-, the same is true for G; that is, dG(vk-1)k and dG(vn-k)n-k. Because kn/2 by (30) and (31), this is contrary to the hypothesis. So we have that the (n+1)-closure H- of G is indeed complete graph and hence that G is Hamilton-connected by (29).

We write Kn-1+e+e for Kn-1 together with a vertex joining two vertices of Kn-1 by edges e,e, respectively.

Lemma 17.

Let G be a connected graph of order n6. If (33)e(G)(n-1)(n-2)2+2, then G is Hamilton-connected unless GKn-1+e+e or GO3K3.

Proof.

Suppose that G is not a Hamilton-connected graph with degree sequence (dG(v1),dG(v2),,dG(vn)), where dG(v1)dG(v2)dG(vn) and n6. By Lemma 16, there is integer kn/2 such that dG(vk-1)k and dG(vn-k)n-k. Since G is connected, k2. Thus, (34)e(G)=12i=1ndG(vi)12[(k-1)k+(n-2k+1)(n-k)+k(n-1)]=12(n2-2nk+3k2-3k+n)=8n2-924+32(k-2n+36)2.

Because 2kn/2, (9-2n)/6k-(2n+3)/6(n-3)/6. Thus, if n6, (35)e(G)8n2-924+32(k-2n+36)2(n-1)(n-2)2+2. Since e(G)(n-1)(n-2)/2+2, then all inequalities in the above argument should be equalities. From the last equality in (35), we have k=2 or k=3 and n=6. If k=2, by the equality in (34), G is a graph with dG(v1)=2,dG(v2)=dG(v3)==dG(vn-2)=n-2,dG(vn-1)=dG(vn)=n-1, which implies GKn-1+e+e. If k=3 and n=6, by the equality in (34), G is a graph with dG(v1)=dG(v2)=dG(v3)=3,dG(v4)=dG(v5)=dG(v6)=5, which implies GO3K3.

Theorem 18.

Let G be a connected graph of order n6. If (36)q(G)2(n-2)+4n-1, then G is Hamilton-connected.

Proof.

By Corollary 11 and (36), we have (37)e(G)q(G)(n-1)-(n-1)(n-2)2(n-1)(n-2)2+2.

Suppose that G is not Hamilton-connected. Then, by Lemma 17 and (37), GKn-1+e+e or GO3K3.

If GKn-1+e+e. Let X=(X1,X2,,Xn)T be the eigenvector of Q(G) corresponding to the eigenvalue q(G). By (36), we know that q(G)n-3 and q(G)n-2. Thus, by (17), all vertices of degree n-2 have the same values given by X, say X1, and all vertices of degree n-1 have the same values, say X2. Denote by X3 the value of the vertex of degree 2 given by X. Also, by (17), we have (38)(q(G)-(n-2))X1=(n-4)X1+2X2,(q(G)-(n-1))X2=(n-3)X1+X2+X3(q(G)-2)X3=2X2. Transform (38) into a matrix equation (B-q(G)I)X=0, where X=(X1,X2,X3)T and (39)B=[2n-620n-3n1022].

Thus, q(G) is the largest root of following equation: (40)q3+(4-3n)q2+(2n2-2n-8)q-4n2+20n-24=0.

Let f(x)=x3+(4-3n)x2+(2n2-2n-8)x-4n2+20n-24; then f(x)=3x2+2(4-3n)x+2n2-2n-8. Let f(x)=0; we have two values x1 and x2, such that f(x1)=f(x2)=0, where (41)x1=3n-4-3n2-18n+403,x2=3n-4+3n2-18n+403.

Hence, f(x) is strictly increasing with respect to x for x>x2.

Consider f(2(n-2)+4/(n-1))=4(n-3)2(n2-3n+6)/(n-1)3>0 and 2(n-2)+4/(n-1)>x2, which implies that q(G)<2(n-2)+4/(n-1).

If GO3K3. We can calculate that q(G)=5+13<8.8=2(6-2)+4/(6-1). Thus, in either case, we have 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 of China under Grant no. 11040606M14, the Natural Science Foundation of Department of Education of Anhui Province of China under Grant nos. KJ2011A195 and KJ2013A196, and the Young Scientific Research Foundation of Anqing Normal College, no. KJ201307.

Fiedler M. Nikiforov V. Spectral radius and Hamiltonicity of graphs Linear Algebra and Its Applications 2010 432 9 2170 2173 10.1016/j.laa.2009.01.005 MR2599851 2-s2.0-77049096567 Zhou B. Signless Laplacian spectral radius and Hamiltonicity Linear Algebra and Its Applications 2010 432 2-3 566 570 10.1016/j.laa.2009.09.004 MR2577702 2-s2.0-70449685722 Lu M. Liu H. Q. Tian F. Spectral radius and Hamiltonian graphs Linear Algebra and Its Applications 2012 437 7 1670 1674 10.1016/j.laa.2012.05.021 MR2946350 2-s2.0-84863990121 Li R. Egienvalues, Laplacian Eigenvalues, and some Hamiltonian properties of graphs Utilitas Mathematica 2012 88 247 257 Butler S. Chung F. Small spectral gap in the combinatorial Laplacian implies Hamiltonian Annals of Combinatorics 2010 13 4 403 412 10.1007/s00026-009-0039-4 MR2581094 ZBLl1229.05193 2-s2.0-75549087109 Fan Y. Z. Yu G. D. Spectral condition for a graph to be Hamiltonian with respect to normalized Laplacian http://arxiv.org/abs/1207.6824 Bondy J. A. Chvatal V. A method in graph theory Discrete Mathematics 1976 15 2 111 135 10.1016/0012-365X(76)90078-9 MR0414429 2-s2.0-0004602289 Hendry G. Extending cycles in bipartite graphs Journal of Combinatorial Theory B 1991 51 2 292 313 10.1016/0095-8956(91)90044-K MR1099078 2-s2.0-1542612554 Chvátal V. On the Hamilton's ideal's Journal of Combinatorial Theory B 1972 12 163 168 10.1016/0095-8956(72)90020-2 So W. Commutativity and spectra of Hermitian matrices Linear Algebra and Its Applications 1994 212-213 15 121 129 Das K. Ch. The Laplacian spectrum of a graph Computers & Mathematics with Applications 2004 48 5-6 715 724 10.1016/j.camwa.2004.05.005 MR2105246 2-s2.0-14744269887 Das K. Ch. Maximizing the sum of the squares of the degrees of a graph Discrete Mathematics 2004 285 1–3 57 66 10.1016/j.disc.2004.04.007 MR2074840 2-s2.0-3142641184 Feng L. H. Yu G. H. On three conjectures involving the signless Laplacian spectral radius of graphs Publications de l'Institut Mathematique 2009 85 99 35 38 10.2298/PIM0999035F MR2536687 2-s2.0-67649232804 Erdös P. Gallai T. On maximal paths and circuits of graphs Acta Mathematica Hungarica 1959 10 3 337 356 Zbl0090.39401