Equienergetic Net-regular Signed Graphs

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The concept of energy is recently generalized to signed graphs i.e. the sum of the absolute values of the eigenvalues of a signed graph Σ. Many authors have constructed non-cospectral equienergeic graphs. In this paper, we established the spectra of heterogeneous unbalanced net-regular signed complete graphs. Then, we give a method to construct an infinite pairs of unbalanced, non-cospectral, equienergetic, net-regular signed graphs of the same order.


Introduction
We consider the graph G is a simple undirected graph without loops and multiple edges with n vertices and m edges.A signed graph (or sigraph ) is an ordered pair Σ = (G, σ), where G = (V, E) is a graph called the underlying graph of Σ and σ : E → {+1, −1}, called signing (or a signature), is a function from the edge set E(G) of G into the set {+1, −1}.If all the edges of Σ are assigned either + or − sign then Σ is known as homogeneous signed graph and heterogeneous otherwise.The sign of a cycle in a sigraph is the product of the signs of its edges.Thus a cycle is positive if it contains an even number of negative edges.A signed graph is said to be balanced (or cycle balanced) if all of its cycles are positive.
Let Σ = (G, σ) be a signed graph with vertex set V (Σ) = V (G) = {v 1 , v 2 , . . ., v n }, then its adjacency matrix is defined as a square matrix A(Σ) = [a ij ] n×n , where The negation of a signed graph Σ = (G, σ), denoted by η(Σ) = (G, σ) is the same graph with all signs reversed.The adjacency matrices are related by The characteristic polynomial of the signed graph Σ is defined as where I is an identity matrix of order n.
The roots of the characteristic equation Φ(Σ : λ) = 0 are called the eigenvalues of signed graph Σ.If the distinct eigenvalues of A(Σ) are λ 1 ≥ λ 2 ≥ • • • ≥ λ n and their multiplicities are m 1 , m 2 , . . ., m n , then the spectrum of Σ is Two signed graphs are cospectral if they have the same spectrum.The spectral criterion for balance in signed graph is given by B.D.Acharya as follows: Theorem 1.1: [1] A signed graph is balanced if and only if it is cospectral with the underlying graph.
The energy of a signed graph [5] is defined as the sum of the absolute values of the eigenvalues of the adjacency matrix A(Σ) of a signed graph Σ, that is, if λ 1 , λ 2 , . . ., λ n are the eigenvalues of Σ, then Two signed graphs Σ 1 and Σ 2 are said to be equienergetic if ε(Σ 1 ) = ε(Σ 2 ).
Let V = {v 1 , v 2 , ...., v n } be the vertex set of a signed graph Σ and d + i (d − i ) be the number of positive(negative) edges incident with v i .However, in signed graph Σ, the degree of v i is defined as sdeg A signed graph Σ is said to be net-regular [9] of degree k if all its vertices have same net-degree equal to k i.e.
It is clear that the net-regularity of a signed graph can be either positive, negative or zero.
In literature, the following results are available to find the eigenvalues of signed graphs.Lemma 1.2: [5] The signed paths P (r) n , where r is the number of negative edges and 0 ≤ r ≤ n − 1, have the eigenvalues(independent of r) given by Lemma 1.3: [14,5] The eigenvalues λ j of signed cycles C n and 0 ≤ r ≤ n are given by where r is the number of negative edges and [r] = 0 if r is even, [r] = 1 if r is odd.
Spectra of graphs is well documented in [4] and signed graphs is discussed in [5,6,7,14].For standard terminology and notations in graph theory we follow D.B.West [18] and for signed graphs we follow T. Zaslavsky [20].
The problem of construction of equienergetic signed graphs is an open problem [5].The main aim of this paper is to construct such signed graphs.First, we establish a spectrum for one class of net-regular signed complete graphs in order to find the spectrum for unbalanced signed graphs on 2n vertices.Then we construct an infinite family of unbalanced non-cospectral and equienergetic net-regular signed graphs on 2n vertices.It is also shown that Coulson's Integral formula for graph energy still holds for signed graphs.

Main Results
Here we established a spectrum for one class of heterogeneous net-regular signed complete graphs.Definition 2.1: Let C n be a cycle on n vertices and C n be its complement where n ≥ 4. Consider a signed graph such that σ(u, v) = 1 if u is adjacent to v in C n and σ(u, v) = -1 if u is adjacent to v in C n .The resultant signed graph is an unbalanced net-regular signed complete graph and we denote it as K net n where n ≥ 4.
Lemma 2.2: [4] Let C n be a cycle on n vertices and C n be its complement.Spectrum of C n is The following result gives the spectrum of unbalanced net-regular signed complete graph K net n .
Theorem 2.3 : The spectrum of heterogeneous unbalanced signed complete graph (K net n ) is Proof : Adjacency matrix of K net n can be written as

Equienergetic Net-regular Signed Graphs
Recently, graph energy is generalized to signed graphs.Here we note that well known Coulson's Integral formula for graphs is valid for signed graphs for calculating the energy and proof of the theorem is analogous to graphs [8].
Theorem 3.1: If Σ is a signed graph then the energy of signed graph Σ is Example 3.2: Let C − 4 be a signed cycle with odd number of negative edges.Then the characteristic polynomial of Now, we shall introduce the following operation to construct heterogeneous, net-regular signed graphs Σ and Σ * .
The following well known result is used for the investigation.

Lemma 3.5 : [4] Let
A 0 A 1 A 1 A 0 be a symmetric 2 × 2 block matrix.Then the spectrum of A is the union of the spectra of A 0 + A 1 and A 0 − A 1 .
Theorem 3.6: Let Σ and Σ * be two net-regular signed graphs as defined above on 2n vertices for n ≥ 4. Then the spectrum of Σ and Σ * are