New Results on Energy of Graphs of Small Order

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1 Global Joural of Pure ad Applied Mathematics. ISSN Volume 13, Number 7 (2017), pp Research Idia Publicatios New Results o Eergy of Graphs of Small Order Sophia Shalii G. B. a* ad Mayamma Joseph b a* Departmet of Mathematics, Mout Carmel College, Bagalore-58, Idia. b Departmet of Mathematics, Christ Uiversity, Bagalore-29, Idia. Abstract The cocept of eergy of graphs was coied by Iva Gutma i The eergy of a graph G is the sum of the absolute values of eigevalues of adjacecy matrix A(G) of the graph G. I this paper we do a detailed study o the bouds of eergy of simple, coected ad udirected graphs of order less tha 10. The problem of fidig bouds of eergy of graphs was addressed by the researchers by obtaiig results i terms of order ad size of the give graph. Our study gives emphasis to bouds of the eergy of a class of graph i terms of eergy of aother class of graph. Idetificatio of Some ew families of hyper-eergetic graphs ad the relatio betwee eergy of wheel graph ad fa graph which is its sub graph also obtaied. It has bee observed that star graphs possess the least eergy amog all the families of graphs uder cosideratio. Also the MATLAB code to geerate the adjacecy matrix ad hece to fid the eergy of certai families of graphs are give. Keywords: Adjacecy matrix; Graph eergy AMS classificatio: 05C50 1. INTRODUCTION A graph ca be represeted i terms of a matrix usig the adjacecy relatio betwee its vertices. Hece it is atural to study the properties of adjacecy matrices associated with graphs. I this directio, the idea of eergy of graphs which is a umber assiged

2 2838 Sophia Shalii G. B. ad Mayamma Joseph to a graph i terms of eigevalues of its adjacecy matrix emerged as a iterestig area of study i graph theory. All graphs cosidered i this paper are simple, fiite ad udirected. Let G be the simple graph with the vertex set V(G) cosistig of vertices labeled by v, v, 1 2 v ad a edge set E(G). The adjacecy matrix A(G) of the graph G is the square matrix of order, whose (i,j) th etry is equal to 1 if the vertices v i ad zero, otherwise. That is aij = 1, if v i ad where aij is ay elemet of A(G). 0, otherwise v j are adjacet v j are adjacet ad is equal to The cocept of eergy was itroduced by Iva Gutma i 1978 [7]. The eergy of a graph is calculated from the eige values of the correspodig adjacecy matrix of the graph. For details ad a overview of the results o graph eergy we refer to the comprehesive work by Li, Shi ad Gutma [8]. The eergy of a graph is the sum of absolute values of the eigevalues of its adjacecy matrix A(G). I other words, if i where i = 1,2, deote the eigevalues of the adjacecy matrix of the graph G of order, the the eergy of the graph deoted by (G) is give by ( G) (1) i1 i Determiig the eergy of differet classes of graphs is a challegig task as it requires the costructio of adjacecy matrix of each ad every graph uder cosideratio. This is oe of the reasos why researchers tried obtaiig bouds for the eergy of graphs. The first boud for graph eergy was by McClellad [1]. He proved that for a graph G of order ad size m, ( G) a 2m, where a is empirical costat ad a 0. 9 ad it is geerally called as McClellad iequality. He also showed that for a graph G with vertices ad m edges 2 m ( G) 2m. The McClellad iequality was improved by Koole, Moulto ad Gutma [9]. They have show that the maximum eergy of a graph of order is 2 / 3. For the graph G with large umber of vertices, Nikifov 2

3 New Results o Eergy of Graphs of Small Order / 3 [11] proved that ( G). The McClellad lower boud iequality was 11/10 2 improved by Kikar Das ad Gutma [10]. From the review of the above results it was observed that ot all the bouds give a correct estimate of the actual value of the eergy of a class of graphs. Motivated by this observatio we have attempted to aswer the followig questios: What are the exact bouds for (G) whe G is a tree of order less tha 10? Is it possible to obtai a formula for the bouds of eergy of a tree of order? It has also bee observed that most of the bouds of eergy of graphs are expressed i terms of the order ad size of the give graph. But there are ot may results where the bouds of eergy of a class of graphs is obtaied i relatio with eergy of some other classes of graphs. I this directio, we explore the followig problem. Give families of graphs G, H, S does there exist ay relatio of the form ( G) ( H) ( S)? Whe we cosider simple graphs o vertices, the complete graph has the maximum umber of edges. Gutma [7] has proved that the eergy of a complete graph o vertices is 2(-1) ad cojectured that all simple graphs of order has eergy less tha 2(-1). However this cojecture was proved to be false [6] leadig to the study of a ew class of graphs called hyper-eergetic graphs. A graph G o vertices is said to be hyper-eergetic if ( G ) 2( 1). Walikar, Ramae ad Hampiholi [6] preseted a systematic costructio of hyper-eergetic graphs. I this paper, we classify the o-hyper-eergetic graphs whe 2 ( 1) m for 10, also, we check are there ay graphs of order less tha 10 other tha complete graphs, complete bipartite graphs whose lie graph is hyper-eergetic? For all the basic otatios ad defiitios we refer the books by West ad Harary [2, 4]. 2. RESULTS AND OBSERVATIONS 2.1 Bouds for Eergy of graphs I this sectio we preset some results o bouds o eergy of plaar graphs i terms of its order ad size. Further we obtai refied bouds for two families of plaar graphs amely wheels ad fas. First we preset the followig theorem which is obtaied usig the property that the umber of edges i ay triagle free plaar graph caot exceed 2 4 [4].

4 2840 Sophia Shalii G. B. ad Mayamma Joseph Theorem 1. If G is a plaar graph of order ad size m ad without triagles, the 2( m 2) ( G) 2 ( 2) Proof. If G is a plaar graph without triagles, the m 2 4. Cosider the McClellad iequality [1], ( G) 2m, Sice m 2 4, we have ( G ) 2(2 4) therefore, ( G ) 2 ( 2) Also ( G ) 2 ( 1) m 4 ( G ) This implies that ( G ) 2( m 2) ( G ) 2 m 2 2 Thus 2( m 2) ( G) 2 ( 2) Amog several families of graphs we cosider wheels ad fa graphs ad study their eergy for the order less tha or equal to 10. The above theorem gives the relatio betwee the eergy of a particular graph with the order ad size of that graph. We try to obtai a relatio betwee eergy of a graph ad eergy of its sub graph. Oe such families of graphs to be are wheel graphs ad fa graphs. We preset the followig MATLAB code to geerate the adjacecy matrix of wheel graphs ad fa graphs ad hece to fid its eergy Matlab code to geerate the eergy of a wheel graph: = ; %eter the umber of vertices A=zeros(); for i=1:-1 A(i,i+1)=1; A(i+1,i)=1; A(i,)=1;

5 New Results o Eergy of Graphs of Small Order 2841 A(,i)=1; ed A(1,-1)=1; A(-1,1)=1; A eigevaluesofwheel=eig(a) eergy=sum(abs(eigevaluesofwheel)) Matlab code to geerate the eergy of a fa graph: = ; %eter the umber of vertices A=zeros(); for i=1:-1 A(i,i+1)=1; A(i+1,i)=1; A(i,)=1; A(,i)=1; ed A eigevaluesoffa=eig(a) eergy=sum(abs(eigevaluesoffa)) Usig the above MATLAB program the eergy of wheel graphs ad fa graphs of order upto 10 are foud show i Table 1.

6 2842 Sophia Shalii G. B. ad Mayamma Joseph Table 1: Eergy of Wheel graphs ad Fa graphs of order up to 10. W W ) ( W W W W W W W F F ) ( F F F F F F F Usig the values from the Table 1, we cojecture the followig: Cojecture 1. If W is the wheel graph with vertices ad m edges, the 2 ( 2) ( W ) 2 ( 2) m Cojecture 2. If F is a fa graph o vertices the ( F ) We also observe that the eergy of fa graph is less tha the eergy of a wheel graph. But we caot geeralize the eergy of a sub graph is always smaller tha the eergy of its origial graph. We observed that most of the bouds are obtaied i terms of the order ad size of the graph. Now we try to fid whether there exist ay relatio betwee the eergies of two differet classes of graphs by aalyzig the eergy of all trees of order less tha Bouds for eergy of trees We ow discuss the results o the eergy of trees of order less tha 10. We have obtaied a iterestig result cocerig with the eergies of star graph S ad the path graph P. Up to order 3, the results o the eergy of path ad the star graphs are trivial. So we cosider graphs of order greater tha 3.

7 New Results o Eergy of Graphs of Small Order Matlab code to geerate the eergy of path: = ; %eter the umber of vertices A=zeros(); for i=1:-1 A(i,i+1)=1; A(i+1,i)=1; ed A eigevaluesofpath=eig(a) eergy=sum(abs(eigevaluesofpath)) Matlab code to geerate the eergy of star: = ; %eter the umber of vertices A=zeros(); for i=1:-1 A(1,i+1)=1; A(i+1,1)=1; ed A eigevaluesofstar=eig(a) eergy=sum(abs(eigevaluesofstar)) Usig the above MATLAB program the eergy of Paths ad Stars of order up to 10 are foud show i Table 2.

8 2844 Sophia Shalii G. B. ad Mayamma Joseph Table 2: Eergy of path graphs ad star graphs of order less tha 10 No.of vertices ( P ) S ) ( From table 2, we observe the followig: Propositio 1. If ( S ) ( P ). S is the star graph ad P is the path graph o vertices, Harary [4] has metioed the umber of graphs o umber of vertices with respect to the umber of edges. Cvetkovic ad Petric [3] have elisted all the coected graphs o 6 vertices. O fidig the eergy of all those graphs, we observe the followig: Propositio 2. If G is a simple coected graph o vertices with =4,5,6 the ( ) ( G). S We ow cosider all the tree graphs of order less tha 10. Harary [4] has elisted all the tree graphs of order up to 10. The eergy of all the tree graphs are listed below: No. of vertices No. of Trees Table 3: Eergy of trees of order less tha 10 Eergy , , , , , 6, ,6.8990, , , ,6.7206, , , , , , 8, , , , , ,7.3846, , , , , ,8.4721, , , , , ,9.3317, , , , , , , , , ,7.8098, , , , , ,8.9981, , , , , ,9.2472, , , , , ,9.4311, , , , , , , , , , , , , , , , , , , , , , ,

9 New Results o Eergy of Graphs of Small Order 2845 From table 3, we obtai a relatio betwee eergy of star graphs ad the eergy of other tree graphs. Cojecture 3. If ( S ) ( T ). S is a star graph of order ad T is tree graph with vertices, the 2.3 Hyper eergetic graphs Theorem 2. If G is a graph with vertices ad T K ) is a total graph of complete ( graph o vertices, the for 3, T K ) is hyper eergetic. Proof. ( The umber of vertices of T ( K ) = umber of vertices of K + umber of edges of K The total graph of = ( 1) ( 1) 2 2 K is isomorphic to the lie graph of 1 K. If G is a o-empty r-regular graph with vertices ad m edges [8], the ( L( G) : ) ( G : 2 r)( 2) m where L (G) is the lie graph of G ad ( G : 2 r) is the polyomial obtaied by replacig with 2 r i ( G : ). The characteristic polyomial of K is ( K : ) ( 1)( 1) 1 Thus, ( L( K ( L( K ) : ) ( 2 4)( 4) 1 ( 2) ( 3) / 2 ( 1)( 2) / 2 1 ) : ) ( 2( 1) 4)( ( 1) 4) ( 2) ( 2 2)( 3) ( 2) ( 1)( 2) / 2

10 2846 Sophia Shalii G. B. ad Mayamma Joseph Sice, T K ) L( K ), we ca write ( L( K ( 1 ) : ) ( 2 2)( 3) ( 2) ( 1)( 2) / 2 Therefore, ( T( K )) 2 2(1) 3( ) 2( 1)( 2) / 2 E To show that G) ( K ), it is sufficiet to show that ( 1) ( Suppose, ( 1), the This holds oly whe 3, a cotradictio to the fact that 3. Hece, ( 1) which completes the proof. We have also observed the followig results: Propositio 4: 1. For 3, the total graph of K, is hyper-eergetic. 2. For 6, the total graph of -cocktail party graph is hyper-eergetic. 3. For 4, the total graph of wheel graph W is hyper-eergetic. 4. For 6 9, the lie graph of wheel graph is hyper-eergetic. Theorem 3. Proof. For 4, all the cycle graphs are o-hyper-eergetic. If G is a k-regular graph o vertices, the ( G) k k( 1)( k) Sice, C is a 2-regular graph, we have ( ) 2 2( 1)( 2) C

11 New Results o Eergy of Graphs of Small Order 2847 I order to show that ( ) 2( 1), it suffices to show that C 2 2( 1)( 2) 2( 1) Suppose, 2 2( 1)( 2) 2( 1) the which is ot possible for 4 Thus, ( C ) 2( 1) for 4 which completes the proof. CONCLUSION Most of the results we have obtaied i this paper is based o the study of the exact values of the eergy of the graphs uder cosideratio. Further studies are to be doe ad appropriate proof techiques are to be employed to geeralize these results. However the results obtaied ca act as a poiter for articulatig ad provig the geeral results. REFERENCES [1] B. McClellad, 1971, Properties of latet roots of a matrix: The estimatio of electro eergies, J. Chem. Phys., vol. 54, pp [2] D.B. West, 2014, Itroductio to Graph Theory, 2d ed., PHI Learig pvt. ltd., New Delhi. [3] D. Cvetkovic, M. Petric, 1984, A table of coected graphs o six vertices, Discrete Math., vol. 50, pp [4] F.Harary, 1970, Graph Theory, Addiso Wesley, Califoria. [5] G. B. Rikma, J. Goedgebeur, 2011, Geeratio of cubic graphs, Discrete Math. Theor. Comput. Sci., vol. 13, pp [6] H.B. Walikar, H. S. Ramae, P. R. Hampiholi, 1998, O the eergy of a graph, Proc. Graph Coectios, pp [7] I. Gutma, 1978, The eergy of a graph, Ber. Math. Stat. Sekt., pp [8] I. Gutma, X. Li, Y. Shi, 2012, Graph Eergy, Spriger. [9] J. H. Koole, V. Moulto, I. Gutma, 2000, Improvig the McClellad iequality for total electro eergy, Chem. Phys. Lett., vol. 320, pp

12 2848 Sophia Shalii G. B. ad Mayamma Joseph [10] K. C. Das, S. A. Mojallal, I. Gutma, 2013, Improvig McClellad lower boud for eergy, MATCH Commu. Math. Comput. Chem., vol. 70, pp [11] V. Nikiforov, 2007, Graphs ad Matrices with maximal eergy, J. Math. Aal. Appl., vol. 327, pp