On the Energy of Some Graphs

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1 nnals of Pure and pplied Mathematics Vol. 7, No., 08, 5- ISSN: X P, online Published on pril 08 DOI: nnals of On the Energy of Some raphs Sudhir R. Jog, Prabhaar R. Hampiholi and njana S. Joshi Department of Mathematics, ogte Institute of Technology Udyambag Belagavi , Karnataa, India Corresponding author Received 8 February 08; accepted pril 08 bstract. Energy of a graph is an interesting parameter related to total π electron energy of the corresponding molecule. Recently Vaidya and Popat defined a pair of new graphs and obtained their energy in terms of the energy of original graph. In this paper we generalize the construction and obtain their energy. lso we discuss the spectrum of the first level thorn graph of a graph. Keywords: Splitting graph, Shadow graph, Thorn graph, Energy of a graph MS Mathematics Subject Classification 00: 05C50. Introduction In quantum chemistry, the seletons of certain non-saturated hydrocarbons are represented by graphs [, 3]. Energy levels of electrons in such a molecule are, in fact, the eigenvalues of the corresponding graph. ll the graphs considered here are finite, simple and undirected. The adjacency matrix of a graph of order n with vertex set {v,v, v n } is an n x n matrix [a ij ] with a ij if v i is adjacent to v j and 0 otherwise. The eigenvalues of are called eigenvalues of. Eigenvalues along with their multiplicities form the spectrum of. We denote spectrum of graph by spec.the addition of absolute eigenvalues of is defined as energy of denoted by E [5]. It is a generalization of a formula valid for the total π-electron energy of a conjugated hydrocarbon as calculated with the Hucel molecular orbital HMO method [-]. For some bounds on the energy of a graph; one can refer [6-8].Two non isomorphic graphs on same number of vertices are said to be cospectral if they have the same spectrum. Two non isomorphic graphs are called equienergetic if they have same energy. Obviously cospectral graphs are equienergetic, so we loo for non isomorphic non cospectral connected graphs which are equienergetic. For wor on equienergetic graphs; one can refer [9-]. For some wor on energy of singular graphs, eccentric energy and degree sum energy; one can refer [3-5]. 5

2 Sudhir R. Jog, Prabhaar R. Hampiholi and njana S. Joshi Let ϵ R m xn and B ϵ R n xp then, the Kronecer product tensor product of and B ab a nb is defined as the matrix B a B a B m mn Proposition.[6]: Let ϵ M m and Bϵ M n.furthermore; if λ is an eigenvalue of with eigenvector x and µ is an eigenvalue of B with eigenvector y then λµ is an eigenvalue of B with eigenvector x y. The present wor generalizes the two graphs defined by Vaidya, Popat [7], so that their results become particular cases.. Energy of splitting graph The splitting graph of a graph was defined in 980 by Sampathumar and Waliar [8]. Here we define the generalized version of the same. Definition.. The splitting graph S of a graph of order n, is obtained from copies of by adding one set of n new vertices corresponding each vertex of say {u,u, u n } and joining each u i to neighbor of v j i for j,,3,,. Here we assume j that v i denotes the i th vertex in j th j l copy of. lso we join every v i to neighbors of v i j for all j,,,. We prove the following. Theorem.. +. Proof: With pertinent labeling of vertices the adjacency matrix of taes the form S O 0 where is adjacency matrix of and O is a zero matrix of order n. the matrix of order + by + coincides with minimum covering 0 matrix of K + complete graph of order + with characteristic polynomial λ - [λ λ ] and eigenvalues 0 - times and ± +/ once see [9]. Using proposition.,we have the spectrum of 6

3 On the Energy of Some raphs n n i λ i λ where λ i i,, n are the eigenvalues of. So that + + ± +. Corollary.3. From equation it s easy to see that if and are two equienergetic graphs, then S and S are equienergetic for all. Corollary.. If in equation we get the splitting graph of and the corresponding result for energy as in [7]. Illustration: If is K and the graph S K is as shown below Figure : The spectrum of K so that spectrum of S K is ± 3 ± 3 ES K Energy of a shadow graph We now define the shadow graph of a graph as follows. Definition 3.. The shadow graph D of a connected graph is constructed by taing copies of. Join each vertex of every copy with the neighbors of corresponding vertex in all remaining - copies. Theorem 3.. E D E. Proof: With pertinent labeling the adjacency matrix of D taes the form D where is the adjacency matrix of.

4 Sudhir R. Jog, Prabhaar R. Hampiholi and njana S. Joshi The matrix of order by, having the characteristic polynomial λ - λ- and hence has eigenvalues 0 - times and once. Using proposition. we have the spectrum of λ i 0 where λ i i,, n n n are the eigenvalues of. Hence, E [ D ]. Corollary 3.. From equation it s easy to see that if and are two equienergetic graphs, then D and D are equienegetic for all. Corollary 3.3. If in equation we get the shadow graph of as defined in [7] and the corresponding result for energy. Illustration: If is K and 3 then is as shown below Figure : The spectrum of K so that spectrum of D 3 K is 3 3. lso ED 3 K First level thorn graphs We consider some class of graphs which are constructed by joining an edge to every vertex and then joining pendent vertices to the end vertices of the edge joined. The graph obtained by this procedure from we call, first level thorn graph denoted by +. graph and first level thorn graph +3 are shown below. a b Figure 3: The spectrum of 0 level thorn graphs is discussed in [0]. In this chapter we compute characteristic polynomial of first level thorn graphs. Lemma.. [] If M is any nonsingular matrix then 8

5 On the Energy of Some raphs Theorem.. The adjacency polynomial of a first level thorn graph of a graph is related with adjacency polynomial of by P +,λ λ n-n λ n P,. Proof: Let + denote the graph obtained from by attaching an edge to each vertex of and then attaching pendent vertices to ends of edges joined. With pertinent labeling the adjacency matrix of + has the form, + where P is a matrix of order n by n with i th column having s in i th row to i th row for i,,3.n, O n denotes a zero matrix and I n denotes the identity matrix of order n. The characteristic equation of + is then λi +. From Lemma. we have λi + λi n M P N M Q Q PM. gain from Lemma. we have λi + 3 Hence the theorem. Corollary.3. From equation 3 the spectrum of + is given by for each eigenvalue of,,,... So that the energy of +, E + where denotes root of the cubic equation ++ 0 for each eigenvalue,,.. of. lso from the cubic equation, it can be observed that if nullity of number of zero eigenvalues is p then + has nullity n n + p and contains + p times in its spectrum. Corollary.. If from equation 3 we get + which is same as the graph with path of length attached to each vertex, having the polynomial N 9

6 Sudhir R. Jog, Prabhaar R. Hampiholi and njana S. Joshi P +, λ λ n P,. Note: from the relations above its clear that, if has singularity p then + will have singularity p and ± times in it s specctrum. n Further if then, + is a unicyclic graph of diameter + having adjacency polynomial P C + n, λ λ n PC n, [ cos ]. Corollary.5. When 0 from equation 3 we have, P +0,λ λ n P, λ coincides with the characteristic polynomial of + in [9] as required. 5. Conclusion The paper deals with general version of splitting graph and shadow graph of a graph; consequently we have graphs, whose energy is positive integral multiple of energy of any given graph. lso we obtained infinite family of equienergetic graphs starting with a pair of equienergetic graphs. lso we give a relation between the characteristic polynomial of first level thorn graph of with that of. cnowledgement. The authors than the anonymous referees for their valuable suggestions and comments which improved the presentation of the wor. REFERENCES. E.Hucel, Quantentheoretische Beitrage zum Benzolproblem, Z. Phys, C..Coulson, On the calculation of the energy in unsaturated hydrocarbon molecules, Proc. Cambridge Phil. Soc., I.utman, The energy of a graph, Ber. Math. Statist. Set. Forschungszenturmraz R.Balarishnan, The energy of a graph, Linear lgebra ppl., Caprossi, D.Cvetovi c, I.utman and B.Hansen, Variable neighborhood graph finding graphs with extreme energy, J. Chem. Inf. Comput. Sci., I.utman, The energy of a graph: old and new results, in lgebraic Combinatorics and pplications,. Betten,. Kohner, R. Laue, and.wassermann, eds., Springer, Berlin, 00, H.Ramane, H.Waliar, S.Rao, B.charya, P.Hampiholi, S.Jog and I.utman, Equienergetic graphs, Kragujevac J. Math., H.Ramane, I.utman, H.Waliar and S.Halarni, nother class of equienergetic graphs, Kragujevac J. Math., Indulal and.vijayumar, Equienergetic self complementary graphs, Czech. Math. J., H.Ramane, H.Waliar, S.Rao, B.charya, P.Hampiholi, S.Jog and I.utman, Spectra and energies of iterated line graphs of regular graphs, ppl. Math. Lett., 8,

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