Bounds of Balanced Laplacian Energy of a Complete Bipartite Graph

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1 Iteratioal Joural of Computatioal Itelligece Research ISSN Volume 13, Number 5 (2017), pp Research Idia Publicatios Bouds of Balaced Laplacia Eergy of a Complete Bipartite Graph K. Ameeal bibi 1, B.Vijayalakshmi 2, R. Jothilakshmi 3 1,2,3 Departmet of Mathematics 1,2 D.K.M College for Wome (Autoomous), Idia 3 Mazharul Uloom College, Idia Abstract Let G be a siged coected graph with order ad size m. The siged laplacia is defied by L = D W, where D is siged degree matrix ad W is a symmetric matrix with zero diagoal etries. The siged laplacia is a symmetric positive semidefiite. Let µ1 µ2...µ-1 µ = 0 be the eige values of the laplacia matrix. The siged laplacia eergy is defied as L E(G) = μ i 2m i=0. I this paper, we defied balaced laplacia eergy of a complete bipartite graph ad its upper bouds are attaied. Keywords: Complete bipartite graph, Balaced graph, Siged Laplacia matrix, Siged laplacia Eergy of a graph. AMS Classificatio: 05C50, 05C69 I INTRODUCTION A Siged graph is a graph with the additioal structure that edges are give a sig of either +1 or -1. Formally, a siged graph is a pair = (Γ,σ) cosistig of a uderlyig graph Γ = (V,E) ad a sigature σ : E {+1,-1}[19,20,21].

2 1158 K. Ameeal bibi, B.Vijayalakshmi ad R. Jothilakshmi Defie the adjacecy matrix A( ) = (aij)x as aij = { σ ij if v i is adjacet to v j 0 otherwise The eigevalues λ1, λ2,...λ, of A, assumed i o icreasig order, are the eigevalues of the graph G. The eergy E(G) of G is defied to be the sum of the absolute values of the eigevalues of G. i.e., E(G) = i=1 λ i [1,2,3,4,12,13]. I.Gutma ad B.zhou defied the Laplacia eergy of a graph G i the year 2006[5,6,8,10,11]. Let G be a fiite, simple ad coected graph with order ad size m. The Laplacia matrix of the graph G deoted by L(G) = D(G) - A(G) is a square matrix of order, where D(G) is the diagoal matrix of vertex degrees of the graph G ad A(G) is the adjacecy matrix. Let µ1, µ2,..., µ form the Laplacia spectrum of its Laplacia matrix G the the Laplacia eergy LE(G) of G is defied as LE(G) = μ i 2m i=0 [14,15,16,17,18]. Defiitios ad Examples: Defiitio 1.1 A Siged graph is just a ordiary graph with each of its edges labelled with either + or a. Defiitio 1.2 Give a siged graph G = (V, W) ( where W is a symmetric matrix with zero diagoal etries), the uderlyig graph of G is the graph with the vertex set V ad the set of (udirected) edges E = { (vi, vj) wij 0}. Defiitio 1.3 Let G be a graph with order ad size m. The Laplacia matrix of the graph G is deoted by L = (Lij) is a square matrix defied by 1 if v i is adjacet to v j L ij = { 0 if v i is ot adjacet to v j d i if v i = v j Where di is the degree of the vertex vi.

3 Bouds of Balaced Laplacia Eergy of a Complete Bipartite Graph 1159 Defiitio 1.4 If (V,W) is a siged graph where W is a (mxm) symmetric matrix with zero diagoal etries ad with the other etries wij ɛ R be arbitrary. The degree of ay vertex vi is defied as d i = d (v i ) = i=0 W ij ad D siged degree matrix where D = diag ( (v1), (v2),... (vm)). Defiitio 1.5 The Siged Laplacia is defied by L = D W, where D is the siged degree matrix. The siged Laplacia is symmetric positive semidefiite. Defiitio 1.6 Let G = (V, W) be a siged graph whose uderlyig graph is coected. The G is balaced if there is a partitio of its vertex set V ito two clusters V1 ad V2 such that all the positive edges coect vertices withi V1 or V2 ad all the egative edges coect vertices betwee V1 ad V2. If the siged graph has eve umber of egative edges the it is called a Balaced Siged graph. Defiitio 1.7 Let µ1, µ2,... µ be the eigevalues of L E (G), which are called Siged Laplacia eigevalues of G. The Siged Laplacia eergy L E(G) of G is defied L E(G) = μ i 2m i=1, where 2m Defiitio 1.8 is the average degree of the graph G. A Complete bipartite graph is a graph whose vertices ca be partitioed ito two vertex subsets V1 ad V2 such that o edge has both ed vertices i the same subset ad every possible edge that could coect vertices i differet subsets is part of the graph. That is, it is a bipartite graph ( V1,V2, E) such that ay two vertices v1, v2 where v1 V1, v2 V2 is a edge i E. A Complete bipartite graph with partitios of size V 1 = m ad V 2 =, is deoted by Km,. A Complete bipartite graph K, is a Circulat graph of order 2 ad size 2. Prelimiaries: I this sectio, we listed some previously kow results that are eeded i the subsequet sectios.

4 1160 K. Ameeal bibi, B.Vijayalakshmi ad R. Jothilakshmi Lemma 1: A Bipartite graph G = (V,E) is a graph whose vertex set V ca be partitioed ito two subsets V1 ad V2 such that every edge of G has oe ed i V1 ad aother ed i V2. If V 1 = V 2, the we say that G is balaced. Lemma2: A Siged graph is balaced if every cycle has a eve umber of egative edges. Lemma 3: A all egative siged graph is balaced if ad oly if it is bipartite. Lemma 4: A Complete bipartite graph Km, is balaced if m =. Lemma 5: Let G be a complete bipartite graph of order 2 with two partite sets V1 ad V2 ad V 1 = V 2 = if ad if oly if d(x) + d(y) = 2 for all x V1 ad y V2. Lemma 6: A Sigig of a graph is a assigmet of weight + 1 or -1 to each of its edges. A cycle C of a siged bipartite graph G is balaced if the sum of the weights of the edges i C is cogruet to 0 ( mod 4). Lemma 7: Let G be a bipartite graph of order with size m. The μ1 4m, with equality holds if ad oly if G is regular. Lemma 8: Let G be a r- regular graph of order. The μi = r λ-i+1, 1 i. Lemma 9: Let G be a graph ad G K. The μ-1 δ, where δ is the miimum degree of G. 2. BALANCED LAPLACIAN ENERGY OF A COMPLETE BIPARTITE GRAPH EXAMPLE 2.1 Let G be a complete bipartite graph K, of order 2 with size 2. The complete bipartite graph cotais two differet vertex subsets. Each set cotais vertices. Each set belogs to oe partitio. The two partitios are coected by egative edges. Each vertex i oe partitio is coected to all other vertices i the secod partitio by meas of a egative edge. So this

5 Bouds of Balaced Laplacia Eergy of a Complete Bipartite Graph 1161 graph cotais a eve umber of egative edges. Therefore this graph G is called a balaced complete bipartite graph. Cosider the complete bipartite graphs K, Figure 1: Complete bipartite graph K, From the figure 1, we obtaied the balaced siged adjacecy matrix of the graph as follows A(K,) = X D(K,) = X2

6 1162 K. Ameeal bibi, B.Vijayalakshmi ad R. Jothilakshmi L (K,) = D(K,) A(K,) L (K,) = X2 The characteristic equatio of L (K,) is (µ +1) -1 (µ 3 +1) = 0. Average degree is 2m = 22 2 = Balaced Laplacia Eergy of complete bipartite graph K, is L E(K,) = 1 (-1) times = = OBSERVATIONS: 3.1. The lower ad upper bouds o the eergy of a bipartite graph G E(G) 4m + ( 2)(detA) 2 (1) E(G) 2m 4m + 2(detA) 2 (2) 3.2. Let G be a bipartite graph of order with size m. The the iequality (1) holds. Equality i (1) is attaied if oly if G K1 (or) G Kp,q (-p-q)k1, p+q, 1 p 2.

7 Bouds of Balaced Laplacia Eergy of a Complete Bipartite Graph Let G be a bipartite graph of eve order with m edges. The the iequality (2) holds. Equality i (2) is attaied if ad oly if G K1 or G 2 K Koole ad Moulto foud the boud for the eergy of bipartite graphs, i terms of ad m ( 2m) E(G) 4m 8m2 + ( 2)(2m ) (3) 2 Equality holds if ad oly if G mk2 ( = 2m) or G Kv,v ( = 2v) Let G be a bipartite graph of order with m edges. The λ 1 = 2m, λ 2 =... = λ 1, λ = 2m if ad oly if G K1 or G mk2 ( = 2m) or G Kv,v ( = 2v). 4. UPPER BOUND OF BALANCED LAPLACIAN ENERGY OF COMPLETE BIPARTITE GRAPH. Theorem 4.1: Let G be a complete bipartite graph of order 2 with m = 2 edges. The L E(G) 2[ + ( 1)(M 2 )] where M = m + M 1 2m2. Equality holds i (3) if ad oly if 2 G K1 K2 or G K1 or G mk2 ( = 2m) or G Kv,v ( = 2v). Proof: We defied γ i = μ i 2m, Sice 2m = d 2 i=1 i, M1(G) = d i 1 2 i=1 μ i = d i (d i + 1). i=1 1 Now, L E(G) = μ 1 + i=2 γ i μ 1 + ( 2) γ i 2 1 i=2 i=1 ad

8 1164 K. Ameeal bibi, B.Vijayalakshmi ad R. Jothilakshmi = 2m + γ 1 + ( 2) (2M 4m2 2 γ i 2 ). Sice G is a complete bipartite graph K, with order 2 ad size m = 2 ad by Lemma 7, γ 1 2m ad by lemma 8, μ-1 δ 2m. Similarly, for μi 2m, we get L E(G) 2[ + ( 1)(M 2 )]. REFERENCES [1] Balakrisha. R, The eergy of a graph, Liear Algebra ad its applicatios(2004) [2] Bapat R. B., Pati S, Eergy of a graph is ever a odd iteger. Bull.Kerala Math. Assoc. 1, (2011). [3] Bo Zhou, Eergy of a graph, MATCH commu. Math. Chem. 51(2004), [4] Bo Zhou ad Iva Gutma, O Laplacia eergy of a graph, Match commu. Math. Comput. Chem. 57(2007) [5] Bo Zhou, More o Eergy ad Laplacia Eergy Math. Commu. Math. Comput. Chem.64(2010) [6] Bo Zhou ad Iva Gutma, O Laplacia eergy of a graph, Liear algebra ad its applicatios, 414(2006) [7] Cvetkovi c.d, Doob M, Sachs H, Spectra of graphs - Theory ad applicatios, Academic Press, New York [8] Fath Tabar G.H, Ashrafi A.R., I. Gutma, Note o Laplacia Eergy of graphs, class Bulleti T(XXXVII de l Academics Serbe des Scieces et des arts(2008). [9] Germia K.A, Shahul Hameed K, Thomas Zaslavsky, O products ad lie graphs, their eigevalues ad eergy, Liear Algebra ad its applicatios 435(2011) [10] Gholam Hossei, Fath - Tabar ad Ali Reza Asharfi, Some remarks o Laplacia eige values ad Laplacia eergy of graphs, Math. Commu., vol.15, No.2, pp (2010). [11] Iva Gutma, Emia Milovaovic, ad Igor Milovaovic - Bouds or Laplacia- Type graph eergies, vol. 16 (2015), No. 1, pp

9 Bouds of Balaced Laplacia Eergy of a Complete Bipartite Graph 1165 [12] Iva Gutma, The eergy of a graph. Ber. Math statist. Sekt. Forschugsz. Graz 103, 1-22(1978). [13] Jua Rada, Atoio Tiio, Upper ad Lower bouds for the eergy of bipartite graph, Joural of Mathematics aalysis ad applicatio, vol.289, pgs (2004). [14] Kikar Ch. Das, Seyed Ahamed Mojallal, Iva Gutma, O eergy ad Laplacia eergy of bipartite graphs, Applied Mathematics ad Computatio 273(2016) [15] Merris.M, Laplacia matrices of graphs, A survey, Li. Algeba Appl (1994) [16] Natha Reff, New Bouds for the Laplacia Spectral Radius of a Siged graph, arxiv:1103, 4629VI[Math.Co] 23 Mar [17] Rajesh kaa.m.r, Dharmedra.B.N, Shashi R ad Ramyashree RA, Maximum degree eergy of certai mesh derived etworks- Iteratioal joural of computer Applicatios, 78 No.8(2013) [18] Rajesh kaa M.R, Dharmedra B.N, ad G. Sridhara, Miimum domiatig eergy of a graph - Iteratioal joural of pure ad Applied Mathematics, 85, No. 4(2013) [19] Yaopig Hou, Jiogsheg Li, ad Yogliag Pa, O the Laplacia eigevalues of siged graphs, Liear ad Multiliear Algebra Vol. 51(2003), [20] Zaslavsky. T, Matrices i the theory of siged graphs. I Iteratioal coferece o Discrete Mathematics (ICDM 2008) ad Graph Theory Day IV (Proc. Lecturer otes), Mysore (2008), pp [21] Zaslavsky.T, A mathematical bibliolography of siged ad gai graphs ad allied areas, VII Editio Electroicc J. Combiatorics 8(1998), Dyamics Surveys,8, pp 124.

10 1166 K. Ameeal bibi, B.Vijayalakshmi ad R. Jothilakshmi