MORE GRAPHS WHOSE ENERGY EXCEEDS THE NUMBER OF VERTICES

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1 Iraia Joural of Mathematical Scieces ad Iformatics Vol. 2, No. 2 (2007), pp MORE GRAPHS WHOSE ENERGY EXCEEDS THE NUMBER OF VERTICES CHANDRASHEKAR ADIGA, ZEYNAB KHOSHBAKHT ad IVAN GUTMAN 1 DEPARTMENT OF STUDIES IN MATHEMATICS, UNIVERSITY OF MYSORE, MANASAGANGOTRI, MYSORE , INDIA FACULTY OF SCIENCE, UNIVERSITY OF KRAGUJEVAC, P. O. BOX 60, KRAGUJEVAC, SERBIA ADIGA C@YAHOO.COM ZNBKHT@GMAIL.COM GUTMAN@KG.AC.YU Abstract. The eergy of a graph G is equal to the sum of the absolute values of the eigevalues of G. Several classes of graphs are kow that satisfy the coditio >, where is the umber of vertices. We ow show that the same property holds for (i) biregular graphs of degree a, b, with q quadragles, if q ab/4 ad 5 a < b (a 1) 2 /2 ; (ii) molecular graphs with m edges ad k pedet vertices, if 6 3 (9m + 2k) m 3 0 ; (iii) triregular graphs of degree 1, a, b that are quadragle free, whose average vertex degree exceeds a, that have ot more tha 12/13 pedet vertices, if 5 a < b (a 1) 2 /2. Keywords: Eergy of graph, Spectral graph theory, Biregular graphs, Triregular graphs Mathematics subject classificatio: Primary 05C50, Secodary 05C90. 1 Correspodig author. Received 10 March 2008; Accepted 20 April 2008 c 2007 Academic Ceter for Educatio, Culture ad Research TMU 57

2 58 CH. ADIGA, Z. KHOSHBAKHT, I. GUTMAN 1. INTRODUCTION I this paper we are cocered with a graph ivariat defied i terms of graph eigevalues. Let G be a simple graph ad let its vertex set be V (G) = {v 1, v 2,..., v }. The adjacecy matrix A(G) of the graph G is a square matrix of order whose (i, j)-etry is equal to uity if the vertices v i ad v j are adjacet, ad is equal to zero otherwise. The eigevalues λ 1, λ 2,..., λ of A(G) are said to be the eigevalues of the graph G, ad are studied withi the Spectral Graph Theory [1]. The eergy of the graph G is defied as E = = λ i. The graph eergy is a ivariat much studied i the mathematical ad mathema-tico chemical literature; for details see the book [8], the reviews [4, 5], ad elsewhere [2, 6, 7, 9, 10, 11, 12, 13, 14]. For the chemical applicatio of E see [5] ad the refereces cited therei. Recetly some classes of graphs satisfyig the iequality (1) > have bee characterized [6, 10]. Amog graphs these are o-sigular graphs, i. e., those for which det A(G) 0 [3]; regular graphs of degree greater tha zero [10]; hexagoal systems (bezeoid graphs) [6]; acyclic molecular graphs, with exactly six exceptios [7, 9, 11]. I this paper we poit out a few other classes of graphs with the same property. I order to do this we first eed some preparatios. 2. DEFINITIONS AND PREVIOUS RESULTS Let, as before, λ 1, λ 2,..., λ be the eigevalues of the graph G. The p-th spectral momet of G is defied as Recall that M p = (λ i ) p. (2) M 2 = ad M 4 = 2 d 2 i + 8q where m ad q are, respectively, the umber of edges ad quadragles i G.

3 MORE GRAPHS WHOSE The followig lower boud for the eergy is kow [12]: M2 3 (3). M 4 For geeralizatios of (3) see [2, 13, 14] Recall that a graph is said to be regular of degree x if all its vertices have the same degree x. A graph G is said to be biregular of degrees x ad y, (x < y), if at least oe vertex of G has degree x ad at least oe vertex has degree y, ad if o vertex of G has degree differet from x or y. The set of all -vertex biregular graphs of degrees x ad y will be deoted by Γ (x, y). For G Γ (a, b), the followig special case of the iequality (3) has bee recetly deduced [6]: (4) (2a + 2b 1) 2ab + 8q. A graph G is said to be triregular of degrees x, y, ad z, (x < y < z), if at least oe vertex of G has degree x, at least oe degree y ad at least oe degree z, ad if o vertex of G has degree differet from x or y or z. The set of all -vertex triregular graphs of degrees x, y, ad z will be deoted by Θ (x, y, z). Withi the theory of graph eergy, i view of its chemical applicatios [5, 8] a coected graph i which there are o vertices of degree greater tha 3 is referred to as a molecular graph. Such graphs represet the carbo atom skeleto of cojugated orgaic molecules, ad play a special role i the Hückel molecular orbital theory. 3. THE MAIN RESULTS Theorem 3.1. Let G Γ (a, b) ad let the umber of quadragles (q) of G be less tha or equal to ab/4. The the iequality (1) holds if 5 a < b (a 1) 2 /2. Proof. We start with the boud (4). If q ab/4, the 2ab + 8q 0. Combiig this with (4) we obtai 1 (2a + 2b 1) i. e., 1 d (2a + 2b 1) > a 1 (2a + 2b 1)

4 60 CH. ADIGA, Z. KHOSHBAKHT, I. GUTMAN where d = / is the average vertex degree of the graph G. But a (2a + 2b 1) 1 if ad oly if (a 1) 2 /2 b. Hece / > 1 if 5 a < b (a 1) 2 /2. Corollary 3.2. Let G Γ (a, b) quadragle free. The (1) holds if 5 a < b (a 1) 2 /2. Cosider ow triregular graphs. If G Θ (1, a, b) ad if G has m edges, the (5) k + a + b = ad (6) 1 k + a a + b b = where k is the umber of vertices of degree 1, whereas a ad b are the umber of vertices of degree a ad b, respectively. From Eqs. (5) ad (6) we get b( k) ( k) a =, b = If d i is the degree of the i-th vertex, the d 2 i = 1 2 k + a 2 a + b 2 b (7) ( k) a( k) [ ] [ ] b( k) ( k) ( k) a( k) = k + a 2 + b 2 = k + ( k)(b + a) ( k)ab. Usig Eq. (7), ad bearig i mid (2), we obtai M 4 = 2k + 2( k)(b + a) 2( k)ab + 8q = (2b + 2a 1)( k) 2ab( k) + k + 8q. Thus the iequality (3) ca be rewritte as (8) (2b + 2a 1)( k) 2ab( k) + k + 8q. If G Θ (1, 2, 3) is a quadragle free molecular graph, the q = 0, a = 2, ad b = 3. From (8) we the have 9( k) 12( k) + k..

5 MORE GRAPHS WHOSE i. e., But m 9m 6 + 2k. m 9m 6 + 2k 1 if ad oly if 6 3 (9m + 2k) m 3 0. Thus we have the followig: Theorem 3.3. Let G be quadragle free molecular graph with vertices, m edges ad k pedet vertices. If 6 3 (9m+2k) 2 +4 m 3 0, the the relatio (1) is satisfied. Corollary 3.4. Let G be quadragle free molecular graph with vertices ad m edges. If 4( 3 + m 3 ) 9m 2, the the relatio (1) is satisfied. Proof. We have k where k is the umber of pedet vertices. If the 9m 2 4( 3 + m 3 ) m m 3 = 6 3 (9m + 2) m (9m + 2k) m 3. Corollary 2 follows ow from the Theorem 2. I coectio with Theorem 2 ad its Corollary 2.1 oe should ote that the case m = 1, i. e., the case whe the molecular graph G is a tree was studied i [7, 9, 11]. It was show there that all acyclic molecular graphs, with exactly six exceptios satisfy the relatio (1). All acyclic molecular graphs with 8 vertices satisfy the relatio (1). Theorem 3.5. Let the graph G Θ (1, a, b) be quadragle free ad let it has ot more tha 12/13 pedet vertices. If 5 a < b (a 1) 2 /2 ad if the average vertex degree of G exceeds a, the the relatio (1) is satisfied. Proof. Sice G is quadragle free, from (8), we have (2b + 2a 1)( k) 2ab( k) + k (9) k (2b + 2a 1)( k) 2ab( k) + k.

6 62 CH. ADIGA, Z. KHOSHBAKHT, I. GUTMAN Note that k 12/13 implies k 12( k) 2ab( k), i. e., 2ab( k) + k 0. Usig this i the iequality (9), we obtai d 1 (2b + 2a 1) > a 2b + 2a 1 where, as before, d = / ad, as required i the statemet of Theorem 3, d > a. Now, a 1 2b + 2a 1 holds if ad oly if (a 1) 2 /2 b. Thus >. Refereces [1] D. Cvetković, M. Doob ad H. Sachs, Spectra of Graphs Theory ad Applicatio, Academic Press, New York, [2] J. A. de la Peña, L. Medoza ad J. Rada, Comparig mometa ad π-electro eergy of bezeoid molecules, Discr. Math., 302 (2005), [3] I. Gutma, Bouds for total π-electro eergy, Chem. Phys. Lett., 24 (1974), [4] I. Gutma, The eergy of a graph: Old ad ew results, i: A. Bette, A. Kohert, R. Laue, A. Wasserma (Eds.), Algebraic Combiatorics ad Applicatios, Spriger Verlag, Berli, (2001), [5] I. Gutma, Topology ad stability of cojugated hydrocarbos. The depedece of total π-electro eergy o molecular topology, J. Serb. Chem. Soc., 70 (2005), [6] I. Gutma, O graphs whose eergy exceeds the umber of vertices, Li. Algebra Appl., i press. [7] I. Gutma, X. Li, Y. Shi ad J. Zhag, Hypoeergetic trees, MATCH Commu. Math. Comput. Chem., 60 (2008), i press. [8] I. Gutma ad O. E. Polasky, Mathematical Cocepts i Orgaic Chemistry, Spriger- Verlag, Berli, [9] I. Gutma ad S. Radeković, Hypoeergetic molecular graphs, Idia J. Chem., 46A (2007), [10] I. Gutma, S. Zare Firoozabadi, J. A. de la Peña ad J. Rada, O the eergy of regular graphs, MATCH Commu. Math. Comput. Chem., 57 (2007), [11] V. Nikiforov, The eergy of C 4 -free graphs of bouded degree, Li. Algebra Appl., i press. [12] J. Rada ad A. Tieo, Upper ad lower bouds for the eergy of bipartite graphs, J. Math. Aal. Appl., 289 (2004), [13] B. Zhou, Lower bouds for eergy of quadragle-free graphs, MATCH Commu. Math. Comput. Chem., 55 (2006), [14] B. Zhou, I. Gutma, J. A. de la Peña, J. Rada ad L. Medoza, O spectral momets ad eergy of graphs, MATCH Commu. Math. Comput. Chem., 57 (2007),