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

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)^t where 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.