BOUNDS FOR THE DISTANCE ENERGY OF A GRAPH

59 Kragujevac J. Math. 31 (2008) 59 68. BOUNDS FOR THE DISTANCE ENERGY OF A GRAPH Harishchadra S. Ramae 1, Deepak S. Revakar 1, Iva Gutma 2, Siddai Bhaskara Rao 3, B. Devadas Acharya 4 ad Haumappa B. Walikar 5 1 Departmet of Mathematics, Gogte Istitute of Techology, Udyambag, Belgaum 590008, Idia (e-mails: hsramae@yahoo.com, revakards@rediffmail.com) 2 Faculty of Sciece, Uiversity of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia (e-mail: gutma@kg.ac.yu) 3 Stat-Math Divisio, Idia Statistical Istitute, 203, Barrackpore Road, Kolkata 700108, Idia (e-mail: raosb@isical.ac.i) 4 Departmet of Sciece ad Techology, Govermet of Idia, Techology Bhava, New Mehrauli Road, New Delhi 110016, Idia (e-mail: acharyad@alpha.ic.i) 5 Departmet of Computer Sciece, Karatak Uiversity, Dharwad 580003, Idia (e-mail: walikarhb@yahoo.co.i) (Received September 17, 2007) Abstract. The distace eergy of a graph G is defied as the sum of the absolute values of the eigevalues of the distace matrix of G. Recetly bouds for the distace eergy of

60 a graph of diameter 2 were determied. I this paper we obtai bouds for the distace eergy of ay coected graph G, thus geeralizig the earlier results. 1. INTRODUCTION I this paper we are cocered with simple graphs, that is graphs without loops, multiple or directed edges. Let G be such a graph, possessig vertices ad m edges. We say that G is a (, m)-graph. Let the graph G be coected ad let its vertices be labelled as v 1, v 2,..., v. The distace matrix of a graph G is defied as a square matrix D = D(G) = [d ij ], where d ij is the distace betwee the vertices v i ad v j i G [3,5]. The eigevalues of the distace matrix D(G) are deoted by µ 1, µ 2,..., µ ad are said to be the D-eigevalues of G. Sice the distace matrix is symmetric, its eigevalues are real ad ca be ordered as µ 1 µ 2 µ. The characteristic polyomial ad eigevalues of the distace matrix of a graph are cosidered i [6 8,14,15,18,36]. The distace eergy E D = E D (G) of a graph G is defied as [18] E D = E D (G) = µ i. The distace eergy is defied i aalogy to the graph eergy [9] E = E(G) = λ i where λ 1, λ 2,..., λ are the eigevalues of the adjacecy matrix A(G) of a graph G [5]. For more results o E(G) see [1,2,4,10 13,16,17,19 25,27 35,37]. If G = K, the complete graph o vertices, the A(K ) = D(K ) ad hece E D (G) = E(G) = 2( 1). I a recet paper [18] Idulal, Gutma, ad Vijaykumar reported lower ad upper bouds for the distace eergy of graphs whose diameter (= maximal distace betwee vertices) does ot exceed two. I this paper we obtai bouds for the distace

61 eergy of arbitrary coected (, m)-graphs, which geeralize the results obtaied i [18]. We first eed the followig Lemma. Lemma 1. Let G be a coected (, m)-graph, ad let µ 1, µ 2,..., µ be its D- eigevalues. The µ i = 0 ad µ 2 i = 2 (1) Proof. µ i = trace[d(g)] = d ii = 0. For i = 1, 2,...,, the (i, i)-etry of [D(G)] 2 is equal to d ij d ji = Hece j=1 j=1 µ 2 i = trace[d(g)] 2 = (d ij ) 2 = 2 j=1 Corollary 1.1 [18]. Let G be a coected (, m)-graph, ad let diam(g) 2, where diam(g) deotes the diameter of a graph G. The µ 2 i = 2[2 2 2 3m]. 2. BOUNDS FOR THE DISTANCE ENERGY Theorem 2. If G is a coected (, m)-graph, the 2 (d ij ) 2 E D (G) 2

62 Proof. Cosider the Cauchy Schwartz iequality ( ) 2 ( ) ( ) a i b i a 2 i b 2 i. Choosig a i = 1 ad b i = µ i, we get ( 2 µ i ) µ 2 i from which E D (G) 2 2 This leads to the upper boud for E D (G). Now ( E D (G) 2 2 = µ i ) µ i 2 = 2 (d ij ) 2 which straightforwardly leads to the lower boud for E D (G). Corollary 2.1. If G is a coected (, m)-graph, the E D (G) ( 1). Proof. Sice d ij 1 for i j ad there are ( 1)/2 pairs of vertices i G, from the lower boud of Theorem 2, E D (G) 2 (d ij ) 2 2 ( 1) 2 = ( 1). Theorem 3. Let G be a coected (, m)-graph ad let be the absolute value of the determiat of the distace matrix D(G). The 2 (d ij ) 2 + ( 1) 2/ E D (G) 2 Proof. I view of Theorem 2, we oly eed to demostrate the validity of the lower boud. This is doe aalogously to the way i which a lower boud for graph eergy is deduced i [26].

63 By defiitio of distace eergy ad Eq. (1) E D (G) 2 = = 2 ( 2 µ i ) = µ 2 i + 2 µ i µ j i i<j (d ij ) 2 + 2 i i<j µ i µ j = 2 ij ) (d 2 + µ i µ j. (2) i j Sice for oegative umbers the arithmetic mea is ot smaller tha the geometric mea, 1 ( 1) µ i µ j i j µ i µ j i j 1/( 1) ( ) 1/( 1) = µ i 2( 1) = µ i 2/ = 2/. (3) Combiig Eqs. (2) ad (3) we arrive at the lower boud. Usig Eq. (1), Corollary 1.1 ad Theorem 3 we have followig result. Corollary 3.1 [18]. Let G be a coected (, m)-graph with diam(g) 2. The 4( 1) 6m + ( 1) /2 E D (G) 2(2 2 2 3m). For a -vertex tree T [3,6], det D(T ) = ( 1) 1 ( 1)2 2 from which we obtai the followig: Corollary 3.2. For a -vertex tree T, 2 (d ij ) 2 + [( 1) +2 4 2 ] 1/ E D (T ) 2

64 Theorem 4. If G is a coected (, m)-graph, the E D (G) 2 2 (d ij ) 2 + ( 1) 2 (d ij ) 2 2 (d ij ) 2. (4) Proof. Our proof follows the ideas of Koole ad Moulto [22,23], who obtaied a aalogous upper boud for ordiary graph eergy E(G). By applyig the Cauchy Schwartz iequality to the two ( 1) vectors (1, 1,..., 1) ad ( µ 2, µ 3,..., µ ), we get 2 ( ( ) µ i ) ( 1) µ 2 i i=2 i=2 (E D (G) µ 1 ) 2 ( 1) 2 (d ij ) 2 µ 2 1 Defie the fuctio E D (G) µ 1 + ( 1) 2 f(x) = x + ( 1) 2 We set µ 1 = x ad bear i mid that µ 1 1. From µ 2 i = 2 (d ij ) 2 (d ij ) 2 µ 2 1. (d ij ) 2 x 2. we get x 2 = µ 2 1 2 (d ij ) 2 i. e. x 2 Now, f (x) = 0 implies x = 2

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