DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 305164 10.1155/2014/305164 305164 Research Article Energy Conditions for Hamiltonicity of Graphs Yu Guidong 1 Cai Gaixiang 1 Ye Miaolin 1 http://orcid.org/0000-0003-3133-7119 Cao Jinde 2,3 Yu Wenwu 1 School of Mathematics & Computation Sciences Anqing Normal College Anqing 246011 China aqtc.edu.cn 2 Department of Mathematics Southeast University Nanjing 210096 China seu.edu.cn 3 Department of Mathematics Faculty of Science King Abdulaziz University Jeddah 21589 Saudi Arabia kau.edu.sa 2014 632014 2014 25 12 2013 22 01 2014 6 3 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.

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 ( G ) = i = 1 n | μ i ( G ) | . Denote by G BPT 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 G BPT having a Hamiltonian cycle in terms of the energy of the quasi-complement of G BPT .

1. Introduction

We consider only undirected simple graphs. Let G = ( V , E ) be a graph of order n with vertex set V = V ( G ) = { v 1 , v 2 , , v n } and edge set E = E ( G ) . We denote by e ( G ) the number of edges of G . The complement of G is denoted by G c : = ( V , E c ) , where E c : = { x y : x V , y V , x y , x y E } . For a bipartite graph G BPT = ( X , Y ; E ) , the quasi-complement of G BPT is denoted by G BPT * : = ( X , Y ; E ) , where E = { x y : x X , y Y , x y E } .

The degree matrix of G is denoted by D ( G ) =    diag ( d G ( v 1 ) , d G ( v 2 ) , , d G ( v n ) ) , where d G ( 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 ) = [ a i j ] of order n , where a i j = 1 if v i is adjacent to v j and a i j = 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 ( G ) = D ( G ) - 1 / 2 L ( G ) D ( G ) - 1 / 2 . Obviously, A ( G ) , Q ( G ) , L ( G ) , and ( 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 μ 1 ( G ) μ 2 ( G ) μ n ( G ) be the eigenvalues of A ( G ) . The energy of G is defined as ( G ) = i = 1 n | μ i ( 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  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  studies the signless Laplacian spectral radius of the complements of a graph and gives conditions for the existence of Hamiltonian path or cycle. Li  establishes sufficient conditions for a graph having Hamiltonian path or cycle in terms of the energy of the graph. Lu et al.  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  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  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) ( G c ) > 2 e ( G c ) - n + 2 - Δ ( G c ) + ( n - 1 ) ( 2 e ( G c ) - ( n - Δ ( G c ) ) 2 ) ,

and n - 2 e ( G c ) Δ ( G c ) n - n - 1 ,

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

and n - 1 - 2 e ( G c ) Δ ( G c ) n - 1 - n - 1 ,

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

and n - 2 - 2 e ( G c ) Δ ( G c ) n - 2 - n - 1 .

Theorem 2.

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

Li  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 ( G c ) > 2 e ( G c ) - 2 n + 2 + e ( G c ) ( n - 1 + 1 ) ,

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

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

Let G B P T = ( X , Y ; E ) be a bipartite graph of order n = 2 r 4 , where | X | = | Y | = r 2 . Then G B P T contains a Hamiltonian cycle, if (5) ( G B P T * ) > 2 e ( G B P T * ) - 2 r + 2 + e ( G B P T * ) ( 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 ) ( 2 e ( G c ) - x 2 ) . If 2 e ( G c ) / n < x < 2 e ( G c ) , (6) f ( x ) = 1 + - ( n - 1 ) x ( n - 1 ) ( 2 e ( G c ) - x 2 ) < 0 . So, f ( x ) is monotonously decreasing when 2 e ( G c ) / n < x 2 e ( G c ) . We notice that if n - 2 e ( G c ) Δ ( G c ) n / 2 , n 3 , we have (7) 2 e ( G c ) n < e ( G c ) n Δ ( G c ) 2 n - Δ ( G c ) 2 e ( G c ) , and Δ ( G c ) n - n - 1 . Hence, when n - 2 e ( G c ) Δ ( G c ) n / 2 , n 3 , we have (8) 2 e ( G c ) - 2 n + 2 + e ( G c ) ( n - 1 + 1 ) = 2 e ( G c ) - 2 n + 2 + f ( e ( G c ) ) 2 e ( G c ) - 2 n + 2 + f ( n - Δ ( G c ) ) = 2 e ( G c ) - n + 2 - Δ ( G c ) + ( n - 1 ) ( 2 e ( G c ) - ( n - Δ ( G c ) ) 2 ) . So Theorem 1(i) improves Theorem 3(i), when n - 2 e ( G c ) Δ ( G c ) n / 2 . By a similar discussion, we have that Theorem 1(ii) improves Theorem 3(ii), when n - 1 - 2 e ( G c ) Δ ( G c ) ( n - 1 ) / 2 .

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

2. Preliminaries

Let G be a graph of order n . Ore  proves that if (12) d G ( u ) + d G ( v ) n - 1 for any pair of nonadjacent vertices u and v , then G has a Hamiltonian path; if (13) d G ( u ) + d G ( v ) n for any pair of nonadjacent vertices u and v , then G has a Hamiltonian cycle. Erdős and Gallai  show that if (14) d G ( u ) + d G ( 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 . For an integer k 0 , the k -closure of a graph   G , denoted by 𝒞 k ( 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 graph   G BPT : = ( X , Y ; E ) , where | X | = | Y | , denoted by 𝒞 k ( G BPT ) , is a graph obtained from G BPT 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 G BPT is unique, independent of the order in which edges are added. We note that d 𝒞 k ( G ) ( u ) + d 𝒞 k ( G ) ( v ) k - 1 for any pair of nonadjacent vertices u and v of 𝒞 k ( G ) , d 𝒞 k ( G BPT ) ( x ) + d 𝒞 k ( G BPT ) ( y ) k - 1 for any pair of nonadjacent vertices x X and y Y of 𝒞 k ( G BPT ) .

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 𝒞 n - 1 ( G ) has a Hamilton path.

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

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

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

A balanced bipartite graph G B P T = ( X , Y ; E ) , where | X | = | Y | = r 2 , has a Hamiltonian cycle, if and only if 𝒞 r + 1 ( G B P T ) 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 ( G ) - 2 ( G - e ) ( G ) + 2 .

For a graph G , let Z ( G ) : = u v E ( G ) ( d G ( u ) + d G ( v ) ) = u V ( G ) d G 2 ( 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 λ i ( M ) be the i th 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) λ i ( B ) + λ j ( C ) λ i + 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) λ 1 ( A ( G ) ) + λ 1 ( D ( G ) ) λ 1 ( Q ( G ) ) . Recalling that λ 1 ( A ( G ) ) = μ ( G ) , λ 1 ( D ( G ) ) = Δ ( G ) , and q ( G ) = λ 1 ( 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 : = 𝒞 n - 1 ( G ) . If H = K n , then the result follows from (12). Suppose that H K n and G does not contain a Hamilton path. Then H does not contain a Hamilton path by Lemma 6(i). We notice that d H ( u ) + d H ( v ) n - 2 for any pair of nonadjacent vertices u and v (always existing) in H . Thus, for any edge u v E ( H c ) , (20) d H c ( u ) + d H c ( v ) = 2 ( n - 1 ) - [ d H ( u ) + d H ( v ) ] n , and Z ( H c ) = u v E ( H c ) [ d H c ( u ) + d H c ( v ) ] n e ( H c ) . By Corollary 12, (21) μ ( H c ) Z ( H c ) e ( H c ) - Δ ( H c ) n - Δ ( H c ) n - Δ ( G c ) .

By Cauchy-Schwartz inequality, we have that (22) ( H c ) = i = 1 n | μ i ( H c ) | = μ ( H c ) + i = 1 n - 1 | μ i ( H c ) | μ ( H c ) + ( n - 1 ) i = 1 n - 1 μ i 2 ( H c ) = μ ( H c ) + ( n - 1 ) ( i = 1 n μ i 2 ( H c ) - μ 2 ( H c ) ) = μ ( H c ) + ( n - 1 ) ( 2 e ( H c ) - μ 2 ( H c ) ) ; the equality holds if and only if | μ 1 ( H c ) | = | μ 2 ( H c ) | = = | μ n - 1 ( H c ) | .

Let f ( x ) = x + ( n - 1 ) ( 2 e ( H c ) - x 2 ) . If 2 e ( H c ) / n < x < 2 e ( H c ) , (23) f ( x ) = 1 + - ( n - 1 ) x ( n - 1 ) ( 2 e ( H c ) - x 2 ) < 0 . So, f ( x ) is monotonously decreasing when 2 e ( H c ) / n < x < 2 e ( H c ) . We find that (24) 2 e ( H c ) n 2 e ( G c ) n Δ ( G c ) < n - Δ ( G c ) μ ( H c ) < 2 e ( H c ) , when Δ ( G c ) n - n - 1 .

Thus (25) ( H c ) f ( μ ( H c ) ) f ( n - Δ ( G c ) ) = n - Δ ( G c ) + ( n - 1 ) ( 2 e ( H c ) - ( n - Δ ( G c ) ) 2 ) .

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 ( G c ) - 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 d H ( u ) + d H ( v ) n - 2 , then e ( H ) ( n - 2 ) + C ( n - 2,2 ) = ( n 2 - 3 n + 2 ) / 2 . Hence s ( n 2 - 3 n + 2 ) / 2 - e ( G ) = ( n 2 - 3 n + 2 ) / 2 - ( C ( n , 2 ) - e ( G c ) ) = e ( G c ) - n + 1 .

By Lemma 8, we have that ( H c ) ( G c ) - 2 s . Thus ( H c ) ( G c ) - 2 e ( G c ) + 2 n - 2 . Because e ( H c ) e ( G c ) , we have that (26) ( G c ) 2 e ( G c ) - n + 2 - Δ ( G c ) + ( n - 1 ) ( 2 e ( G c ) - ( n - Δ ( G c ) ) 2 ) , a contradiction.

(ii) Let H - = 𝒞 n ( G ) . If H - = K n , then the result follows from (13). Suppose that H - K n 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 d H - c ( u ) + d H - c ( v ) n - 1 for any edge u v E ( H - c ) and (27) Z ( H - c ) = u v E ( H - c ) [ d H - c ( u ) + d H - 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 ( G c ) .

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

Let s : = e ( H - ) - e ( G ) . Then e ( G c ) - 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 d H - ( u ) + d H - ( v ) n - 1 , then e ( H - ) ( n - 1 ) + C ( n - 2,2 ) = ( n 2 - 3 n + 4 ) / 2 . Hence s ( n 2 - 3 n + 4 ) / 2 - e ( G ) = ( n 2 - 3 n + 4 ) / 2 - ( C ( n , 2 ) - e ( G c ) ) = e ( G c ) - n + 2 .

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

(iii) Let H ^ = 𝒞 n + 1 ( G ) . If H ^ = K n , then the result follows from (14). Suppose that H ^ K n 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 d H ^ c ( u ) + d H ^ c ( v ) n - 2 for any edge u v E ( H ^ c ) , and (32) Z ( H ^ c ) = u v E ( H ^ c ) [ d H ^ c ( u ) + d H ^ 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 ( G c ) .

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

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

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

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

Suppose that G BPT does not contain a Hamilton cycle. Then H BPT : = 𝒞 r + 1 ( G BPT ) does not contain a Hamilton cycle by Lemma 7. Then H BPT is not K r , r by (13). Observe that d H BPT ( x ) + d H BPT ( y ) r for any pair of nonadjacent vertices x X and y Y (always existing) in H BPT . Thus, (37) d H BPT * ( x ) + d H BPT * ( y ) = r - d H BPT ( x ) + r - d H BPT ( y ) r , for any pair of adjacent vertices x X and y Y in H BPT * . Then (38) Z ( H BPT * ) = u V ( H BPT * ) d H BPT * 2 ( u ) = x y E ( H BPT * ) ( d H BPT * ( x ) + d H BPT * ( y ) ) r e ( H BPT * ) . By Corollary 12, we have that (39) μ ( H BPT * ) Z ( H BPT * ) e ( H BPT * ) - Δ ( H BPT * ) r - Δ ( H BPT * ) r - Δ ( G BPT * ) .

Since H BPT * is a bipartite graph, μ n ( H BPT * ) = - μ 1 ( H BPT * ) . By Cauchy-Schwartz inequality, we have (40) ( H BPT * ) = i = 1 n | μ i ( H BPT * ) |    = 2 μ ( H BPT * ) + i = 2 n - 1 | μ i ( H BPT * ) | 2 μ ( H BPT * ) + ( n - 2 ) i = 2 n - 1 μ i 2 ( H BPT * ) = 2 μ ( H BPT * ) + ( n - 2 ) ( i = 1 n μ i 2 ( H BPT * ) - 2 μ 2 ( H BPT * ) ) = 2 μ ( H BPT * ) + ( n - 2 ) ( 2 e ( H BPT * ) - 2 μ 2 ( H BPT * ) ) ; the equality holds if and only if | μ 2 ( H BPT * ) | = = | μ n - 1 ( H BPT * ) | .

Let g ( x ) = 2 x + ( n - 2 ) ( 2 e ( H BPT * ) - 2 x 2 ) . If 2 e ( H BPT * ) / n < x < e ( H BPT * ) , (41) g ( x ) = 2 + - 2 ( n - 2 ) x ( n - 2 ) ( 2 e ( H BPT * ) - 2 x 2 ) < 0 . So, g ( x ) is monotonously decreasing when 2 e ( H BPT * ) / n < x < e ( H BPT * ) . We find that (42) 2 e ( H BPT * ) n Δ ( G BPT * ) < r - Δ ( G BPT * ) μ ( H BPT * ) < e ( H BPT * ) , when Δ ( G c ) r - r . So (43) ( H BPT * ) g ( μ ( H BPT * ) ) g ( r - Δ ( G BPT * ) ) = 2 ( r - Δ ( G BPT * ) ) + ( n - 2 ) ( 2 e ( H BPT * ) - 2 ( r - Δ ( G BPT * ) ) 2 ) . Let t : = e ( H BPT ) - e ( G BPT ) . Then e ( G BPT * ) - e ( H BPT * ) = ( r 2 - e ( G BPT ) ) - ( r 2 - e ( H BPT ) ) = t . Because H BPT has two nonadjacent vertices x X and y Y such that d H BPT ( x ) + d H BPT ( y ) r , so e ( H BPT ) ( r - 1 ) 2 + r = r 2 - r + 1 . Hence t r 2 - r + 1 - e ( G BPT ) = r 2 - r + 1 - ( r 2 - e ( G BPT * ) ) = e ( G BPT * ) - r + 1 .

By Lemma 8, we have that ( H BPT * ) ( G BPT * ) - 2 t . Then ( H BPT * ) ( G BPT * ) - 2 e ( G BPT * ) + 2 r - 2 . Because e ( H BPT * ) e ( G BPT * ) , we have that (44) ( G BPT * ) 2 e ( G BPT * ) + 2 - 2 Δ ( G BPT * ) + ( n - 2 ) ( 2 e ( G BPT * ) - 2 ( r - Δ ( G BPT * ) ) 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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