Signless Laplacian Spectral Conditions for Hamiltonicity of Graphs

K n a complete graph of order n, O n an empty graph of order n (without edges), and K n,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


Introduction
Let a graph,  = (, ) be a simple graph of order  with vertex set  = {V 1 , V 2 , . . ., V  } and edge set . Denote by () := || the number of edges of the graph .Write by   a complete graph of order ,   an empty graph of order  (without edges), and  , a complete bipartite graph with two parts having ,  vertices, respectively.The graph  is said to be Hamiltonian, if it has a Hamiltonian cycle which is a cycle of order  contained in .The graph  is said to be traceable if it has a Hamiltonian path which is a path of order  contained in .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  is defined to be a matrix () = [  ] × , where   = 1 if V  is adjacent to V  and   = 0 otherwise.The largest eigenvalue of () is called to be the spectral radius of , which is denoted by ().The degree matrix of  is written by () = diag(  (V 1 ),   (V 2 ), . . .,   (V  )), where   (V  ) ( = 1, 2, . . ., ) denotes the degree of the vertex V  in the graph .The signless Laplacian matrix of  is defined by () = () + ().The largest eigenvalue of () is called to be the signless Laplacian spectral radius of , which is denoted by ().
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 [1][2][3][4][5][6].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.

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 (), and let Δ() be maximum degree of .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 [2]).Let  be a graph with at least one edge.Then, if and only if  is regular or semi-regular.
Li [4] has given a sufficient condition for a bipartite graph to be Hamiltonian as follows.
The join of  and , denoted by  ∨ , is the graph obtained from ∪ by adding edges joining every vertex of  to every vertex of .
Let  be a graph containing a vertex V.
Lemma 10 (see [12]).Let G be a connected graph.Then with equality if and only if  is a regular graph or a semi-regular graph.
In fact, if  is disconnected, there exists a component  of  such that So the inequality ( 14) also holds when  is a disconnected graph.By Lemmas 9 and 10, we have the following result; also see [13].
Corollary 11.Let  be a graph of order .Then, If  is connected, then the equality in (16) holds if and only if  =  1,−1 or  =   .Otherwise, the equality in (16) holds if and only if  =  −1 + V.
Given a graph  of order , a vector  ∈ R  is called a function defined on , if there is a 1-1 map  from () to the entries of , simply written as   = () for each  ∈ (),   is also called the value of  given by .If  is an eigenvector of () corresponding to the eigenvalue , then  is defined naturally on ; that is,   is the entry of  corresponding to the vertex .One can find that where   (V) denotes the neighborhood of V in .The equation ( 17) is called (, )-eigenequation of .
Theorem 12. Let  be a connected graph of order  ≥ 4. If then  is traceable.
Thus, in either case, we have a contradiction.
Lu et al. [3] have given a sufficient condition for a graph to be traceable as follows.
So, we can apply Theorem 12 but not Theorem 13 for  to be traceable.

Signless Laplacian Spectral Radius in Hamilton-Connected Graphs
For a graph  of order , Erdös and Gallai [14] prove that if for any pair of nonadjacent vertices  and V, then  is Hamilton-connected.The idea for the closure of a graph can be found in [7].For a positive integer , the -closure of a graph  = (, ), denoted by C  (), is a graph obtained from  by successively joining pairs of nonadjacent vertices  ∈  and V ∈ , whose degree sum is at least  until no such pairs remain.By the definition of the -closure of , we have that  C  () () +  C  () (V) ≤  − 1 for any pair of nonadjacent vertices  ∈  and V ∈  of C  ().
Lemma 15 (see [7]).Let  be a graph of order .Then,  is Hamilton-connected if and only if C +1 () is Hamiltonconnected.
Proof.Let  = C +1 () be the ( + 1)-closure of .Next, we will prove that  is a complete graph; then the result follows according to (29).To the contrary, suppose that  is not a complete graph, and let  and V be two nonadjacent vertices in  with and   ()+  (V) being as large as possible.By the definition of C +1 (), we have Then  has at least  − 1 vertices of degree not exceeding  and at least  −  vertices of degree not exceeding  − .Because  is a spanning subgraph of , the same is true for ; that is,   (V −1 ) ≤  and   (V − ) ≤  − .Because  ≤ /2 by (30) and (31), this is contrary to the hypothesis.So we have that the ( + 1)-closure  of  is indeed complete graph and hence that  is Hamilton-connected by (29).