On the Laplacian Energy of Windmill Graph and Graph D {M

International Journal of Contemporary Mathematical Sciences Vol. 11, 2016, no. 9, 405 - 414 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2016.6844

On the Laplacian Energy of Windmill Graph

and Graph Dm,Cn

Minhui Zhao and Zhiping Wang

Department of Mathematics, Dalian Maritime University 116026-Dalian, P.R. China

Copyright © 2016 Minhui Zhao and Zhiping Wang. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let G be a graph with n vertices and m edges, and also let µi(i = 1, 2, ··· , n) be the eigenvalues of the Laplacian matrix of the graph G. The Laplacian energy of the graph G is deﬁned as n X 2m LE = LE(G) = µi − n i=1 In this paper,we present some upper bounds of the Laplacian energy for special

graphs, which are the windmill graph and the graph Dm,Cn . The bounds of the Laplacian energy are given by the methods of partitioning of matrice.

Keywords: Laplacian energy, windmill graph, graph Dm,Cn

1 Introduction

Let G = (V,E) be a simple graph with vertex set V (G) = {ν1, ν2 ··· νn} and edge set E(G). Let A(G) be the (0, 1)-adjacency matrix of G and D(G) be the diagonal matrix of vertex 406 Minhui Zhao and Zhiping Wang degrees. The Laplacian matrix of G is L(G) = D(G) − A(G). The Laplacian matrix has nonnegative eigenvalues, which are denoted by µ1, µ2, ··· µn, arranged in a non-increasing order, n ≥ µ1 ≥ µ2 ≥ µ3 · · · ≥ µn = 0 [3, 4, 5]. When we consideration more than one graph, then we write µi(G) instead of µi. Let di be the degree of the vertex vi for i = 1, 2, ..., n. The maximum vertex degree is denoted by ∆. The intention for Laplacian energy stems from graph energy E(G) [2]. The energy E(G) of a graph G is equal to the sum of the absolute values of the eigenvalues of the adjacency matrix of G. We have recently proposed an energy-like quantity LE(G), based on the eigenvalues of the Laplacian matrix of G. The Laplacian energy of the graph G is deﬁned as [4, 5] n X 2m LE = LE(G) = µi − n i=1 Laplacian energy is a broad measure of graph complexity. Song et al [11] have introduced component-wise Laplacian graph energy, as a complexity measure useful to ﬁlter image descrip- tion hierarchies. This paper is organized as followed. In section 2 , we start with showing some preliminary lemmas and theorems relies on the pigeonhole principle. Then, In section 3-4, we distinguish calculate the Laplacian energy and the upper bound of the Laplacian energy of above special graphs, which are given by the methods of partitioning of matrices [12].

2 Preliminaries

In this section, we shall list some previously known results that will be needed in the next two sections. The eigenvalues of a real matrix X are the square roots of the eigenvalues of the matrix XXt, where Xt represent the transpose of the matrix X. Then for the graph G with n Pn vertices, we have LE(G) = i=1 µi(L(G) − d(G)En). Lemma2.1 ([7,9]) Let X and Y be two n × n real matrices. Then

n n X X µi(X + Y ) ≤ µi(X) + µi(Y ). (1) i=1 i=1 Lemma2.2 ([8]) Let G be a graph on n vertices which has at least one edge. Then the Laplacian eigenvalues are labeled so that µ1 ≥ µ2 ≥ · · · µn,then

µ1 ≥ ∆ + 1. (2)

Lemma 2.3 ([1]) Let G be a graph on n vertices with Laplacian spectrum µ1, µ2, ··· , µn−1, µn = 0 Then n X LE(G) ≤ max{µi, d(G)}. (3) i=1 On the Laplacian energy of Windmill graph 407

(m) Figure 1: The windmill graph of K3

(m) 3 The Windmill Graph Kn

(m) The windmill graph Kn [6] are deﬁned as only one centre vertex, which included m of the complete graph Kn. The number of vertices are N = m(n − 1) + 1, and the number of edges mn(n−1) mn(n−1) are M = 2 , the average of degree is d = m(n−1)+1 . (m) The principle of labeling vertices of the windmill graph Kn is the ﬁrst of the centre vertex is labeled, the following each complete graph is labeled in proper sequence. See Figure1. By massive calculation, we discover the following rules. The eigenvalues of Laplacian have below regulation.

(a)µ1 = m(n − 1) + 1, µm(n−1)+1 = 0. (b) The number of eigenvalues of 1 is equal to m − 1, the remaining eigenvalues are equal to n. (m) Theorem 3.1 The Laplacian energy of windmill graph Kn is given by

(m) 2 LE(Kn ) = 2 − mn − 2n + n m + d(3m − nm − 1). 408 Minhui Zhao and Zhiping Wang

Proof First we have to prove that

m(n−1)+1 (m) X LE(Kn ) = |µi − d| i=1 m(n−1)+1−m m(n−1)+1−m m(n−1) m(n−1) X X X X = µi − d + d − µi i=1 i=1 i=m(n−1)+2−m i=m(n−1)+2−m m(n−1)+1−m m(n−1)+1−m m(n−1)+1 m(n−1) X X X X = µ1 + µi − d + d − µi i=2 i=1 i=m(n−1)+2−m i=m(n−1)+2−m By the equality (a)(b), we have

= m(n − 1) + 1 + (mn − 2m)n − (mn − 2m + 1)d + md − (m − 1) = m(n − 2) + 2 + n(nm − 2m) + d(3m − mn − 1) = 2 − mn − 2n + n2m + d(3m − nm − 1).

This completes the proof. (m) Theorem 3.2 The upper bound of Laplacian energy of the windmill graph Kn is given by

(m) 2 LE(Kn ) ≤ mn + m(d + n − 1) + 1.

Proof From the above equality (a)(b) and the equality (3), we have

m(n−1)+1 (m) X m LE(Kn ) ≤ maxµi, d(Kn ) i=1 mn−2m+1 m(n−1)+1 X X ≤ µi + d i=1 i=mn−2m+2 m(n−1)+1−m m(n−1)+1 X X ≤ µ1 + µi + d i=2 i=m(n−1)+2−m ≤ mn2 + m(d + n − 1) + 1.

This completes the proof.

(m) Theorem 3.3 Let Kn is a windmill graph, then the upper bound of Laplacian energy is given by

(m) p LE(Kn ) ≤ 2 n(n − 1) + (n − 1)(n − 1 − m) + nd + (m − 1)(n − 1)(d + n − 1).

Proof Note that ! D (K(m)) − d(K(m))E − A(K ) B L(K(m)) − d(K(m))E = n n n n n n n m(n−1)+1 t (m) B R(Kn ) On the Laplacian energy of Windmill graph 409

since   n − 1 ······ 0  . . .   . .. ··· .  t   BB =  . . .   . ··· .. .    0 ······ 0 n×n The Essay [11] had introduced the inequation, so we get v v u   u   u n − 1 ······ 0 u n − 1 ······ 0 u u n u  . .. .  u n  . .. .  X u  . . ··· .  u X  . . ··· .  p uµi   ≤ un µi   = n(n − 1). u  . .. .  u  . .. .  i=1 u  . ··· . .  u i=1  . ··· . .  t   t   0 ······ 0 0 ······ 0

As we are knowed, the part of the Lapalcian matrix is

(2) ! (m)) R(Kn ) 0 R(Kn = (m−1) . 0 R(Kn )

And

 (2)  n − 1 − d(Kn ) · · · −1 (2)  .  R(Kn ) =  −1 .. −1    (2) −1 ··· n − 1 − d(Kn )

This is ,

4 The Graph Dm,Cn

The graph Dm,Cn [10] consists of m cycles with one common vertex, which denoted by v1. And each cycle has n vertices besides the center point v1. So the number of vertices are 410 Minhui Zhao and Zhiping Wang

Figure 2: The graph Dm,C6

2mn (n − 1)m + 1 and mn edges. Then the average of degree is d = (n−1)m+1 .

The principle of labelling vertices of the Dm,Cn graph is the ﬁrst of the common vertex is labeled, the following each cycle is labelled in proper sequence. See Figure 2.

Theorem 4.1 The upper bound of the Laplacian energy of the Dm,Cn graph is given by

p LE(Dm,C2 ) ≤ 2(m + n − 1) + nd + 2 n(m − 1) + (d − 2)(m − 1), m ≥ 2. p LE(Dm,Cn ) ≤ 2(m + n − 1) + 2 2n(m − 1) + (n − 1)(m − 1)(2 + d) + nd, m ≥ 3, n ≥ 3.

Proof. According to the summary , we found the Laplacian matrix of graph of D2,C2 , we may see

! D2(D2,C2 ) + {−d(D2,C2 )E2 − M(D2,C2 ) P L(D2,C ) − d(D2,C )E(n−1)m+1 = (4) 2 2 P t 2

Which On the Laplacian energy of Windmill graph 411

! 0 −1 M = −1 0 ! 1 0 PP t = 0 0

We conclude the Laplacian matrix of the graph of Dm,C2 , using the above result, we get

0 ! D (D ) + {−d(D )E − M(D ) P L(D ) − d(D )E = n m,C2 m,C2 n m,C2 m,C2 m,C2 (n−1)m+1 0t P W (Dm,C2 ) Which ! W (Dm−1,C2 ) 0 W (Dm,C ) = 2 0 2   ! m − 1 ··· 0 0 0t 1 0  . . .  P P =  . .. .  0 0   0 ··· 0

It is known W (D2,C2 ) = 2. We get p LE(Dm,C2 ≤ 2(m + n − 1) + nd + 2 n(m − 1) + (d − 2)(m − 1), m ≥ 2.

First part of the proof is done. Now we suppose that the equality holds in (4). Then we continue found the relationship with the Laplacian matrix of the D2,Cn , (n ≥ 3), we found the following the expressions.

00 ! D (D ) + {−d(D )E − M(D ) P L(D ) − d(D )E = n 2,Cn 2,Cn n 2,Cn 2,Cn 2,Cn (n−1)m+1 00t P W(m−1)(n−1)(D2,Cn ) Which, ! W (D2,Cn−1 ) X(n−2)×1 W (D2,Cn ) = t X1×(n−2) 2     2(m − 1) ··· 0 0 ··· 0 00 00t  . . .  t  . .  P P =  . .. .  X(n−2)×1X =  . .. 0    1×(n−2)   0 ··· 0 0 ··· 1 n×n

Thus we explore that the matrix of W (D2,Cn ) is a symmetric matrix. Merely be divided by us to research more better, which by partitioning of matrix. So in fact

(n−1)(m−1) (n−1)(m−1) X X |µi(W (D2,Cn ))| = |µi(D(n−1)(m−1)(D2,Cn ) − d(D2,Cn )E(n−1)(m−1))| i=1 i=1 412 Minhui Zhao and Zhiping Wang

Using the results, we get the Laplacian matrix of the graph Dm,Cn (m ≥ 3, n ≥ 3) is given by

000 ! Dn(Dm,Cn ) + {−d(Dm,Cn )En − M(Dm,Cn ) P L(Dm,C ) − d(Dm,C )E(n−1)m+1 = 000 (5) n n P t W (Dm,Cn Which   ! 2(m − 1) ··· 0 W (D ) 0 000 000 m−1,Cn t  . .. .  W (Dm,Cn ) = P P =  . . .  0 w(Dm−1,Cn )   0 ··· 0

Although the matrix of M(Dm,Cn ) are diﬀerent, but all of the M(Dm,Cn ) are symmetric matrix.

And the element of the M(Dm,Cn ) are 0 and -1. So by contradiction, we get n n n X X X |µi[Dn(Dm,Cn ) + {−d(Dm,Cn )En − M(Dm,Cn )}]| ≤ |µi[Dn(Dm,Cn )]| + |µi[−d(Dm,Cn En]| i=1 i=1 i=1 Then we have

n 000 ! n X 0 P X LE(D ) ≤ µ ( ) + |µ (D (D ) + {−d(D )E − M(D )}| m,Cn i 000t i n m,Cn m,Cn n m,Cn i=1 P 0 i=1 m(n−1)+1−n X + |µi(W (Dm,Cn )|. i=1 p LE(Dm,Cn ) ≤ 2(m + n − 1) + 2 2n(m − 1) + (n − 1)(2 + d)(m − 1) + nd, m ≥ 3, n ≥ 3.

Consequently, the above conclusions have a permit. This completes the proof.

5 Further Problems

This paper calculate the upper bound for the Laplacian energy of these special graphs, It is easy to verify for derivative graph of complete graph Kn, and the method of partitioning of matrice is used to calculate. It is likely to conjecture the other graphs whether can be used. It would be interesting to study whether the block graphs combined together through a common vertex have similar conclusions or analogous conclusions.

Acknowledgements. The work was supported by the Dalian Science and Technology Project Under contract No. 2015A11GX016 and Fundamental Research Funds for the Central Universities No. 3132016304. On the Laplacian energy of Windmill graph 413

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Received: August 28, 2016; Published: October 15, 2016