On the Laplacian Energy of Windmill Graph. and Graph D m,cn

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1 International Journal of Contemporary Mathematical Sciences Vol. 11, 2016, no. 9, HIKARI Ltd, On the Laplacian Energy of Windmill Graph and Graph D m,cn Minhui Zhao and Zhiping Wang Department of Mathematics, Dalian Maritime University 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 µ i 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 LE = LEG = n µ i 2m n In this paper,we present some upper bounds of the Laplacian energy for special graphs, which are the windmill graph and the graph D m,cn. The bounds of the Laplacian energy are given by the methods of partitioning of matrice. Keywords: Laplacian energy, windmill graph, graph D m,cn 1 Introduction Let G = V, E be a simple graph with vertex set V G = {ν 1, ν 2 ν n } and edge set EG. Let AG be the 0, 1-adjacency matrix of G and DG be the diagonal matrix of vertex

2 406 Minhui Zhao and Zhiping Wang degrees. The Laplacian matrix of G is LG = DG AG. The Laplacian matrix has nonnegative eigenvalues, which are denoted by µ 1, µ 2, µ n, arranged in a non-increasing order, n µ 1 µ 2 µ 3 µ n = 0 [3, 4, 5]. When we consideration more than one graph, then we write µ i G instead of µ i. Let d i be the degree of the vertex v i for i = 1, 2,..., n. The maximum vertex degree is denoted by. The intention for Laplacian energy stems from graph energy EG [2]. The energy EG 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 LEG, based on the eigenvalues of the Laplacian matrix of G. The Laplacian energy of the graph G is defined as [4, 5] n LE = LEG = µ i 2m n 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 description 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 XX t, where X t represent the transpose of the matrix X. Then for the graph G with n vertices, we have LEG = n µ ilg dge n. Lemma2.1 [7,9] Let X and Y be two n n real matrices. Then n µ i X + Y n µ i X + µ i Y. 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 µ 1 µ 2 µ n,then µ Lemma 2.3 [1] Let G be a graph on n vertices with Laplacian spectrum µ 1, µ 2,, µ n 1, µ n = 0 Then LEG n max{µ i, dg}. 3

3 On the Laplacian energy of Windmill graph 407 Figure 1: The windmill graph of K m 3 3 The Windmill Graph K m n The windmill graph K n m [6] are defined as only one centre vertex, which included m of the complete graph K n. The number of vertices are N = mn 1 + 1, and the number of edges are M = mnn 1 2, the average of degree is d = mnn 1 mn 1+1. The principle of labeling vertices of the windmill graph K n m 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. aµ 1 = mn 1 + 1, µ mn 1+1 = 0. b The number of eigenvalues of 1 is equal to m 1, the remaining eigenvalues are equal to n. Theorem 3.1 The Laplacian energy of windmill graph K n m is given by LEK m n = 2 mn 2n + n 2 m + d3m nm 1.

4 408 Minhui Zhao and Zhiping Wang Proof First we have to prove that LEK m n = = mn 1+1 mn 1+1 m µ i d mn 1+1 m µ i d + mn 1+1 m mn 1+1 m = µ 1 + µ i d + i=2 By the equality ab, we have mn 1 i=mn 1+2 m d mn 1+1 i=mn 1+2 m mn 1 i=mn 1+2 m d µ i mn 1 i=mn 1+2 m = mn mn 2mn mn 2m + 1d + md m 1 = mn nnm 2m + d3m mn 1 = 2 mn 2n + n 2 m + d3m nm 1. This completes the proof. Theorem 3.2 The upper bound of Laplacian energy of the windmill graph K n m is given by LEK m n mn 2 + md + n Proof From the above equality ab and the equality 3, we have This completes the proof. LEK m n mn 1+1 mn 2m+1 maxµ i, dk m n µ i + mn 1+1 i=mn 2m+2 mn 1+1 m µ 1 + µ i + i=2 mn 2 + md + n d mn 1+1 i=mn 1+2 m d µ i Theorem 3.3 Let K n m given by is a windmill graph, then the upper bound of Laplacian energy is LEK m n 2 nn 1 + n 1n 1 m + nd + m 1n 1d + n 1. Proof Note that LK n m dk n m E mn 1+1 = D n K n m dk n m E n AK n B B t RK n m

5 On the Laplacian energy of Windmill graph 409 since n 1 0. BB t.... = n n The Essay [11] had introduced the inequation, so we get n 1 0 n. n n. µ i n µ i = nn 1. As we are knowed, the part of the Lapalcian matrix is RK m n = RK 2 n 0 0 RK m 1 n. And RK n 2 = n 1 dk n n 1 dk n 2 This is, LEK m n + + n µ i n n This completes the proof. 0 B B t 0 µ i D n K n m dk n m E n AK n µ i RK m n 2 n 1 + n 1n 1 + m + nd + m 1n 1d n The Graph D m,cn The graph D m,cn [10] consists of m cycles with one common vertex, which denoted by v 1. And each cycle has n vertices besides the center point v 1. So the number of vertices are

6 410 Minhui Zhao and Zhiping Wang Figure 2: The graph D m,c6 n 1m + 1 and mn edges. Then the average of degree is d = 2mn n 1m+1. The principle of labelling vertices of the D m,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 D m,cn graph is given by LED m,c2 2m + n 1 + nd + 2 nm 1 + d 2m 1, m 2. LED m,cn 2m + n nm 1 + n 1m 12 + d + nd, m 3, n 3. Proof. According to the summary, we found the Laplacian matrix of graph of D 2,C2, we may see LD 2,C2 dd 2,C2 E n 1m+1 = D 2 D 2,C2 + { dd 2,C2 E 2 MD 2,C2 P P t 2 4 Which

7 On the Laplacian energy of Windmill graph M = 1 0 P P t 1 0 = 0 0 We conclude the Laplacian matrix of the graph of D m,c2, using the above result, we get D n D m,c2 + { dd m,c2 E n MD m,c2 P LD m,c2 dd m,c2 E n 1m+1 = P t W D m,c2 Which W D m,c2 = P P t = W D m 1,C m It is known W D 2,C2 = 2. We get LED m,c2 2m + n 1 + nd + 2 nm 1 + d 2m 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 D 2,Cn, n 3, we found the following the expressions. D n D 2,Cn + { dd 2,Cn E n MD 2,Cn P LD 2,Cn dd 2,Cn E n 1m+1 = P t W m 1n 1 D 2,Cn Which, W D2,Cn 1 W D 2,Cn = X n 2 1 X1 n 2 t 2 2m P P t =..... X n 2 1X1 n 2 t = Thus we explore that the matrix of W D 2,Cn is a symmetric matrix. Merely be divided by us to research more better, which by partitioning of matrix. So in fact n 1m 1 µ i W D 2,Cn = n 1m 1 n n µ i D n 1m 1 D 2,Cn dd 2,Cn E n 1m 1

8 412 Minhui Zhao and Zhiping Wang Using the results, we get the Laplacian matrix of the graph D m,cn m 3, n 3 is given by D n D m,cn + { dd m,cn E n MD m,cn P LD m,cn dd m,cn E n 1m+1 = P t W Dm,Cn 5 Which W D m,cn = W D m 1,Cn 0 0 wd m 1,Cn 2m 1 0 P P t = Although the matrix of MD m,cn are different, but all of the MD m,cn are symmetric matrix. And the element of the MD m,cn are 0 and -1. So by contradiction, we get n µ i [D n D m,cn + { dd m,cn E n MD m,cn }] Then we have n 0 P LED m,cn µ i + P t 0 n µ i [D n D m,cn ] + n µ i [ dd m,cn E n ] n µ i D n D m,cn + { dd m,cn E n MD m,cn } mn 1+1 n + µ i W D m,cn. LED m,cn 2m + n nm 1 + n 12 + dm 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 K n, 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

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10 414 Minhui Zhao and Zhiping Wang in image hierarchies using Laplacian graph energy, Chapter in: Computer Vision-ECCV 2010, Part IV, K. Daniilidis, P. Maragos, N. Paragios Eds., Springer, Berlin, 2010, [12] Z.Z. Liu, Energy, Laplacian Energy and Zagreb Index of Line Graph, Middle Graph and Total Graph, Int. J. Contemp. Math. Sciences, , no. 18, Received: August 28, 2016; Published: October 15, 2016