Orderenergetic, hypoenergetic and equienergetic graphs resulting from some graph operations

A graph G is said to be orderenergetic, if its energy equal to its order and it is said to be hypoenergetic if its energy less than its order. Two non-isomorphic graphs of same order are said to be equienergetic if their energies are equal. In this paper, we construct some new families of orderenergetic graphs, hypoenergetic graphs, equienergetic graphs, equiorderenergetic graphs and equihypoenergetic graphs.


Introduction
In this paper, we consider simple undirected graphs. Let G = (V, E) be a simple graph of order p and size q with vertex set V (G) = {v 1 , v 2 , ..., v p } and edge set E(G) = {e 1 , e 2 , ..., e q }. The adjacency matrix A(G) = [a ij ] of the graph G is a square symmetric matrix of order p whose (i, j) th entry is defined by a i,j = 1, if v i and v j are adjacent, 0, otherwise .
The eigenvalues λ 1 , λ 2 , ..., λ p of the graph G are defined as the eigenvalues of its adjacency matrix A(G). If λ 1 , λ 2 , ..., λ t are the distinct eigenvalues of G, then the spectrum of G can be written as spec(G) = λ 1 λ 2 ... λ t m 1 m 2 ... m t , where m j indicates the algebraic multiplicity of the eigenvalue λ j , 1 ≤ j ≤ t of G. The energy [7] of the graph G is defined as ε(G) = p i=1 |λ i |. More results on graph energy are reported in [2,7].
Two non-isomorphic graphs are said to be cospetral if they have the same spectrum, otherwise they are known as non-cospectral. Two non-isomorphic graphs of the same order are said to be equienergetic if they have the same energy [12]. A graph of order p is said to be hyperenergetic if its energy is greater than 2(p − 1), otherwise graph is non hyperenergetic. Graphs of order p with energy equal to p is called orderenergetic graphs [1]. The number of graphs whose energy equal to its order are relatively small. So we are trying to find new families of orderenergetic graphs. The orderenergetic graphs are studied in [1]. The spectrum of complete bipartite graph K p,p is Then ε(K p,p ) = 2p, so K p,p is orderenergetic for every p. So our interest is to find the orderenergetic graphs other than K p,p . In 2007, I.Gutman et al. [10] introduced the definition of hypoenergetic graphs. A graph is said to be hypoenergetic if its energy is less than its order, otherwise it is said to be non hypoenergetic. The properties of hypoenergetic graphs are discussed in detail [6,9,10]. In the chemical literature there are many graphs for which the energy exceeds the order of graphs. In 1973, England and Ruedenberg published a paper [5] in which they asked "why does the graph energy exceed the number of vertices?". In 2007, Gutman [8] had proved that if the graph G is regular of any non-zero degree, then G is non hypoenergetic. The orderenergetic and hypoenergetic graphs have several applications in theoretical chemistry. A graph is said to be integral if all of its eigenvalues are integers. The aim of this paper is to construct new families of orderenergetic, hypoenergetic and equienergetic graphs using some graph operations.
The complement graph G of G is a graph with vertex set same as that of G and two vertices in G are adjacent only if they are not adjacent in G. We shall use the following notations throughout this paper, C p , K p , P m and K r,s denotes cycle on p vertices, complete graph on p vertices, path on m vertices and complete bipartite graph on r +s vertices respectively. The symbols I m and J m will stands for the identity matrix of order m and m × m matrix with all entries are ones respectively.
The rest of the paper is organized as follows. In Section 2, we state some previously known results that will be needed in the subsequent sections. In Section 3, we construct some orderenergetic graphs. In Section 4, some new families of hypoenergetic graphs are presented. In Section 5, an infinite family of equienergetic, equiorderenergetic and equihypoenergetic graphs are given.

Preliminaries
In this section, we recall the concepts of the m-splitting graph, the m-shadow graph and the m-duplicate graph of a graph and list some previously established results.
Definition 2.1. [4] The Kronecker product of two graphs G 1 and G 2 is a graph G 1 ×G 2 with vertex set V (G 1 ) × V (G 2 ) and the vertices (x 1 , x 2 ) and (y 1 , y 2 ) are adjacent if and only if (x 1 , y 1 ) and (x 2 , y 2 ) are edges in G 1 and G 2 respectively.

Definition 2.2. [4]
Let A ∈ M m×n (R) and B ∈ M p×q (R) be two matrices of order m × n and p × q respectively. Then the Kronecker product of A and B is defined as follows be two matrices of order n. Let λ be an eigenvalue of matrix A with corresponding eigenvector x and µ be an eigenvalue of matrix B with corresponding eigenvector y, then λµ is an eigenvalue of A ⊗ B with corresponding eigenvector x ⊗ y.
Lemma 2.1. [3]. If G 1 and G 2 are any two graphs, then ε( Proposition 2.2. [4]. If G 1 is a r 1 -regular graph with n 1 vertices and G 2 is a r 2regular graph with n 2 vertices, then the characteristic polynomial of G 1 ∨ G 2 is given by Note that the number of vertices in D m (G) is pm.
If m = 2, then the graph D 2 (G) is called shadow graph of G.
Definition 2.6. [13]. Let G = (V, E) be a (p, q) graph with vertex set V and edge set E. Let W be a set such that V W = ∅, |V | = |W | and f : V → W be bijective (for a ∈ V we write f (a) as a for convenience ). A duplicate graph of G is

Construction of orderenergetic graphs
In this section, it is possible to construct an infinite family of orderenergetic graphs from the given orderenergetic graphs. Let G and H be orderenergetic graphs, then G ∪ H is orderenergetic. For example, the graph K p,p ∪ mK 2 is orderenergetic, but this graph is not connected.

Hypoenergetic graphs
In 2007, I.Gutman et al. [10] introduced the definition of hypoenergetic graphs. In this section, we present some techniques for constructing sequence of hypoenergetic graphs.
Proposition 4.1. Kronecker product of two hypoenergetic graphs is hypoenergetic.
The following theorem enable us to construct infinitely many hypoenergetic graphs.
Proposition 4.2. Let G 1 be an orderenergetic graph and G 2 be a hypoenergetic graph. Then Kronecker product of G 1 and G 2 , G 1 × G 2 is hypoenergetic graph .
Let G be a graph of order p and we denote G s r = K r,s × G, r, s ∈ N . It is very interesting to construct hypoenergetic graphs from non hypoenergetic graphs.
The following theorem describes a construction of hypoenergetic graphs from the complete graph. Proof. The energy of complete graph is 2(p − 1) and |V (G m 1 )| = p(1 + m). As m ≥ 14, Thus G m 1 is hypoenergetic whenever m ≥ 14.
Remark 4.2. The graph G = K p is non hypoenergetic but G m 1 is hypoenergetic for m ≥ 14.

Equienergetic graphs
In this section, we construct some new pairs of equienergetic graphs.
Proposition 5.1. Let G be a (p, q) graph. Then the graphs D m (D(G)) and D 2m (G) are non-cospectral equienergetic graphs.
Proposition 5.2. Let G be a (p, q) graph and D m (G) be the m-duplicate graph of G. Then ε(D m (G)) = 2 m ε(G).  The following propositions describes the class of equiorderenergetic graphs.
Proposition 5.6. Let G be a hypoenergetic graph. Then the graphs spl 2 (G) and D 3 (G) are equihypoenergetic graphs.
Conclusion In this paper, we construct some family of orderenergetic graphs from the known orderenergetic graphs. Also, some new families of hypoenergetic graphs are derived by using some graph operations. Moreover, the problem for constructing equienergetic graphs are discussed. In addition to that a new class of equiorderenergetic and equihypoenergetic graphs are obtained.