ON EQUIENERGETIC, HYPERENERGETIC AND HYPOENERGETIC GRAPHS

The eigenvalue of a graph G is the eigenvalue of its adjacency matrix and the energy E(G) is the sum of absolute values of eigenvalues of graph G. Two non-isomorphic graphs G1 and G2 of the same order are said to be equienergetic if E(G1) = E(G2). The graphs whose energy is greater than that of complete graph are called hyperenergetic and the graphs whose energy is less than that of its order are called hypoenergetic graphs. The natural question arises: Are there any pairs of equienergetic graphs which are also hyperenergetic (hypoenergetic)? We have found an affirmative answer of this question and contribute some new results.


Introduction
We begin with finite connected and undirected graphs without loops and multiple edges. The terms not defined here are used in sense of Balakrishnan and Ranganathan [1] or Cvetković et al. [5]. The adjacency matrix of a graph G with vertices v 1 , v 2 , . . . , v n is an n × n matrix [a ij ] such that, The eigenvalues of adjacency matrix of graph is known as eigenvalues of graph. The set of eigenvalues of the graph with their multiplicities is known as spectrum of the graph. Hence, spec(G) = λ 1 λ 2 · · · λ n m 1 m 2 · · · m n .
Two non-isomorphic graphs are said to be cospectral if they have same spectra, otherwise they are known as non-cospectral. Let G be a graph on n vertices and λ 1 , λ 2 , . . . , λ n be the eigenvalues of G. The energy of a graph G is the sum of absolute values of the eigenvalues of graph G and denoted by E(G). Hence, The concept of energy was introduced by Gutman [6]. A brief account of energy of graph can be found in Cvetković et al. [5] and Li et al. [10]. Two non-isomorphic graphs G 1 and G 2 of same order are said to be equienergetic if E(G 1 ) = E(G 2 ).
Ramane et al. [12,13] have proved that if G 1 and G 2 are regular graphs of same order then for k ≥ 2, L k (G 1 ) and L k (G 2 ), L k (G 1 ) and L k (G 2 ) are equienergetic. Here, L k (G) is called iterated line graph of G.
Some equienergetic graphs have been described in Li et al. [10], while a symmetric computer aided study have carried out for equienergetic trees [2,11]. Some open problem on equienergetic graphs were posted in [8]. To find out non-copspectral equienergetic graphs other than trees is challenging and interesting as well. We take up this problems and construct a pair of graphs which are equienergetic.
In 1978 Gutman [6] conjectured that among all graphs with n vertices, the complete graph K n has the maximum energy. This was disproved by Walikar et al. [16] and was defined the concept of hyperenergetic graphs whose energy is greater than that of complete graphs. Gutman [7] has proved that hyperenergetic graphs on n vertices exist for all n ≥ 8 and there are no hyperenergetic graphs on less than 8 vertices.
A graph G on order n is said to be hypoenergetic [3] if E(G) is less than its order otherwise it is said to be non-hypoenergetic [4]. In 2007 Gutman [9] have proved that if the graph G is regular of any non-zero degree, then G is non hypoenergetic.
The present work is aimed to contribute to find families of hyperenergetic and hypoenergetic.
The splitting graph S (G) of a graph G is obtained by adding to each vertex v a new vertex v , such that v is adjacent to every vertex that is adjacent to v in G. The shadow graph D 2 (G) of a connected graph G is constructed by taking two copies of G say G and G . Join each vertex u in G to the neighbors of the corresponding vertex u in G . Vaidya and Popat [15] have proved that for any graph G, The m-shadow graph D m (G) of a connected graph G is constructed by taking m copies of G, say G 1 , G 2 , . . . , G m , then join each vertex u in G i to the neighbors of the corresponding vertex v in G j , 1 ≤ i, j ≤ m.

Equienergetic Graphs
Proof. Let G be any graph with n vertices. Then, D 3 (G) and Spl 2 (G) are graphs with 3n vertices. According to Proposition 1.1 and Proposition 1.2,  Proof. Consider a complete graph K n on n vertices. Then, S (K n ) is a graph with 2n vertices. It is obvious that energy of complete graph with 2n vertices is 2(2n − 1).   Here, The following is a graph of E(S (K n )) and E(K 2n ) which helps to understand that S (K n ) is hyperenergetic when n ≥ 6.

Figure 3
The natural question arises: Are there any graphs which are equienergetic and hyperenergetic as well? To answer this question we prove following corollary. Corollary 3.1. D 3 (S (K n )) and Spl 2 (S (K n )) are equihyperenergetic graphs for n ≥ 9.
Proof. As we have discussed in Theorem 3.1, S (K n ) is a graph with 2n vertices. Therefore, D 3 (S (K n )) is a graph with 6n vertices. To prove above result we show that D 3 (S (K n )) is hyperenergetic if and only if n ≥ 9.