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

A signed graph is a pair that consists of a graph and a sign mapping called signature from to the sign group . In this paper, we discuss the -path product signed graph where vertex set of is the same as that of and two vertices are adjacent if there is a path of length , between them in the signed graph . The sign of an edge in the -path product signed graph is determined by the product of marks of the vertices in the signed graph , 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 which are switching equivalent to -path product signed graphs for which are switching equivalent to and also the negation of the signed graph ŋ that are switching equivalent to for . We also characterize signed graphs that are switching equivalent to -distance signed graph for where 2-distance signed graph defined as follows: the vertex set is same as the original signed graph and two vertices , are adjacent if and only if there exists a distance of length two in . The edge is negative if and only if all the edges, in all the distances of length two in are negative otherwise the edge is positive. The -path network along with these characterizations can be used to develop model for the study of various real life problems communication networks.•-path product signed graph.•-distance signed graph.