Harary energy of complement of line graphs of regular graphs

. The Harary matrix of a graph G is de(cid:133)ned as H ( G ) = [ h ij ] , where h ij = 1 d ( v i ;v j ) , if i 6 = j and h ij = 0 , otherwise, where d ( v i ;v j ) is the distance between the vertices v i and v j in G . The H -energy of G is de(cid:133)ned as the sum of the absolute values of the eigenvalues of Harary matrix. Two graphs are said to be H -equienergetic if they have same H -energy. In this paper we obtain the H -energy of the complement of line graphs of certain regular graphs in terms of the order and regularity of a graph and thus constructs pairs of H -equienergetic graphs of same order and having di⁄erent H -eigenvalues.


Introduction
Let G be a simple, undirected, connected graph with n vertices and m edges. Let the vertices of G be labeled as v 1 ; v 2 ; : : : ; v n . The adjacency matrix of a graph G is the square matrix A(G) = [a ij ] , in which a ij = 1 if v i is adjacent to v j and a ij = 0 , otherwise. The eigenvalues of A(G) are the adjacency eigenvalues of G , and they are labeled as 1 2 n . These form the adjacency spectrum of G [4]. Two graphs are said to be cospectral if they have same spectra.
The distance between the vertices v i and v j , denoted by d(v i ; v j ), is the length of the shortest path joining v i and v j . The diameter of a graph G, denoted by diam(G) , is the maximum distance between any pair of vertices of G. A graph G is said to be r-regular graph if all of its vertices have same degree equal to r.
The Harary matrix was used in the study of molecules in the quantitative structure property relationship (QSPR) models [9].
The Harary index de…ned as the sum of the reciprocal of the distances between all pairs of vertices and it can be derived from the Harary matrix. It has interesting properties in structure-property correlations [11,16].
The eigenvalues of H(G) labeled as 1 2 n are said to be the Harary eigenvalues or H-eigenvalues of G and their collection is called Harary spectrum or H-spectrum of G. Two non-isomorphic graphs are said to be H-cospectral if they have same H-spectra.
The Harary energy or H-energy of a graph G, denoted by HE(G), is de…ned as [5] The Harary energy is de…ned in full analogy with the ordinary graph energy E(G) , de…ned as [6] The ordinary graph energy has a relation with the total -electron energy of a molecule in quantum chemistry [10]. Bounds for the Harary energy of a graph are reported in [3,5].
Two connected graphs G 1 and G 2 are said to be Harary equienergetic or Hequienergetic if HE(G 1 ) = HE(G 2 ) . The H-equienergetic graphs are reported in [12,13]. The distance energy of complements of iterated line graphs of regular graphs has been obtained in [8]. In this paper we use similar technique of [8] to obtain the H-energy of the complement of line graphs of certain regular graphs and thus construct H-equienergetic graphs having di¤erent H-spectra.
The complement of a graph G is a graph G, with vertex set same as of G and two vertices in G are adjacent if and only if they are not adjacent in G. The line graph of G, denoted by L(G) is the graph whose vertices corresponds to the edges of G and two vertices of L(G) are adjacent if and only if the corresponding edges are adjacent in G. For k = 1; 2; : : : the k-th iterated line graph of G is de…ned as If G is a regular graph of order n 0 and of degree r 0 then the line graph L(G) is a regular graph of order n 1 = (n 0 r 0 )=2 and of degree r 1 = 2r 0 2. Consequently the order and degree of L k (G) are [1,2] n k = r k 1 n k 1 2 and r k = 2r k 1 2; (4) where n i and r i stands for order and degree of L i (G), i = 0; 1; : : :. Therefore r k = 2 k r 0 2 k+1 + 2 (5) and We need following results.
If G is an r-regular graph, then its maximum adjacency eigenvalue is equal to r.
Since r All adjacency eigenvalues of a regular graph of degree r satisfy the condition r i r [4]. Therefore i + r 0 , i = 1; 2; : : : ; n . Therefore by (9), Hence r k 1 n k 1 1 2 : Therefore by Theorem 6, HE L k (G) = HE L(L k 1 (G)) = r k 1 (n k 1 2): Corollary 9. Let G be a regular graph of order n 0 and of degree r 0 . Let n k and r k be the order and degree respectively of the k-th iterated line graph L k (G), k 1.

H-equienergetic graphs
If G 1 and G 2 are the regular graphs of same order and of same degree. Then L(G 1 ) and L(G 2 ) are of the same order and of same degree. Further their complements are also of same order and of same degree. Lemma 10. Let G 1 and G 2 be regular graphs of the same order n and of the same degree r. If r n 1 2 , then L(G 1 ) and L(G 2 ) are H-cospectral if and only if G 1 and G 2 are cospectral.
Proof. Follows from Eqs. (7), (8) and (9). Lemma 11. Let G 1 and G 2 be regular graphs of the same order n and of the same degree r. If r n 1 2 , then for k 1, L k (G 1 ) and L k (G 2 ) are H-cospectral if and only if G 1 and G 2 are cospectral.
Theorem 12. Let G 1 and G 2 be regular, non H-cospectral graphs of the same order n and of the same degree r. If r n 1 2 , then for k 1, L k (G 1 ) and L k (G 2 ) form a pair of non H-cospectral, H-equienergetic graphs of equal order and of equal number of edges.
Proof. Follows from Lemma 11 and Corollary 9.