On Net-Regular Signed Graphs

A signed graph is an ordered pair Σ = (G,σ), where G = (V,E) is the underlying graph of Σ and σ : E → {+1,−1}, called signing function from the edge set E(G) of G into the set {+1,−1}. It is said to be homogeneous if its edges are all positive or negative otherwise it is heterogeneous. Signed graph is balanced if all of its cycles are balanced otherwise unbalanced. It is said to be net-regular of degree k if all its vertices have same net-degree k i.e. k = d± Σ(v) = d+ Σ(v) − d− Σ(v), where d+ Σ(v)(d− Σ(v)) is the number of positive(negative) edges incident with a vertex v. In this paper, we obtained the characterization of net-regular signed graphs and also established the spectrum for one class of heterogeneous unbalanced net-regular signed complete graphs