Energy Conditions for Hamiltonicity of Graphs

Let G be an undirected simple graph of order n. Let A(G) be the adjacency matrix of G, and let μ 1 (G) ≤ μ 2 (G) ≤ ⋅ ⋅ ⋅ ≤ μ n (G) be its eigenvalues. The energy of G is defined as E(G) = ∑n i=1 |μ i (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 quasicomplement of GBPT.

The degree matrix of  is denoted by () = diag (  (V 1 ),   (V 2 ), . . .,   (V  )), where   (V) denotes the degree of a vertex V in the graph .We denote by Δ() the maximum degree of .The adjacency matrix of  is defined by the matrix () = [  ] of order , where   = 1 if V  is adjacent to V  and   = 0 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 L() = () −1/2 ()() −1/2 .Obviously, (), (), (), and L() 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  1 () ≤  2 () ≤ ⋅ ⋅ ⋅ ≤   () be the eigenvalues of ().The energy of  is defined as E() = ∑  =1 |  ()|.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 1 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 Hamiltonconnected 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.
Li [3] has given some energy conditions for a graph having Hamiltonian paths or cycles as follows.

Preliminaries
Let  be a graph of order .Ore [7] proves that if for any pair of nonadjacent vertices  and V, then  has a Hamiltonian path; if for any pair of nonadjacent vertices  and V, then  has a Hamiltonian cycle.Erdős and Gallai [8] show that if for any pair of nonadjacent vertices  and V, then  is Hamilton-connected.
The idea for the closure of a graph  was given by Bondy and Chvátal [9].For an integer  ≥ 0, the -closure of a graph , denoted by C  (), 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  BPT := (, ; ), where || = ||, denoted by C  ( BPT ), is a graph obtained from  BPT 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  BPT is unique, independent of the order in which edges are added.We note that  C  () () +  C  () (V) ≤  − 1 for any pair of nonadjacent vertices  and V of C  (),  C  ( BPT ) () +  C  ( BPT ) () ≤  − 1 for any pair of nonadjacent vertices  ∈  and  ∈  of C  ( BPT ).
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 , 1 ≤  ≤ .

Corollary 12.
Let  be a graph with at least one edge.Then
Because  has at least two nonadjacent vertices  and V such that By Lemma 8, we have that (ii) Let  = C  ().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    () +    (V) ≥  − 1 for any edge V ∈ (  ) and By Corollary 12, We find that when Δ(  ) ≤  − 1 − √  − 1.Using similar arguments as in the proof of Theorem 1(i), we have that have that a contradiction.

Proof of Theorem 2
Proof.Suppose that  BPT does not contain a Hamilton cycle.Then  BPT := C +1 ( BPT ) does not contain a Hamilton cycle by Lemma 7. Then  BPT is not  , by (13).Observe that   BPT () +   BPT () ≤  for any pair of nonadjacent vertices  ∈  and  ∈  (always existing) in  BPT .Thus, for any pair of adjacent vertices  ∈  and  ∈  in  * BPT .Then