On derived t-path, t=2,3 signed graph and t-distance signed graph

Abstract

A signed graph Σ is a pair Σ=(Σu,σ)that consists of a graph (Σu,E) and a sign mapping called signature σ from E to the sign group {+,−}. In this paper, we discuss the t-path product signed graph (Σ)^twhere vertex set of (Σ)^t is the same as that of Σ and two vertices are adjacent if there is a path of length t, between them in the signed graph Σ. The sign of an edge in the t-path product signed graph is determined by the product of marks of the vertices in the signed graph Σ, where the mark of a vertex is the product of signs of all edges incident to it. In this paper, we provide a characterization of Σ which are switching equivalent to t-path product signed graphs (Σ)^t for t=2,3 which are switching equivalent to Σ and also the negation of the signed graph ŋ(Σ) that are switching equivalent to (Σ)^t for t=2,3. We also characterize signed graphs that are switching equivalent to t-distance signed graph (Σ¯)t for t=2 where 2-distance signed graph (Σ¯)2=(V′,E′,σ′) defined as follows: the vertex set is same as the original signed graph Σ and two vertices u,v∈(Σ¯)2, are adjacent if and only if there exists a distance of length two in Σ. The edge uv∈(Σ¯)2 is negative if and only if all the edges, in all the distances of length two in Σ are negative otherwise the edge is positive. The t-path network along with these characterizations can be used to develop model for the study of various real life problems communication networks. • t-path product signed graph. • t-distance signed graph.

Keywords

• t-path product and t-distance signed graph