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 E to the sign group $$ . In this paper, we discuss the t-path product signed graph $$ where vertex set of $$ is the same as that of $$ and two vertices are adjacent if there is a path of length t, between them in the signed graph $$ . The sign of an edge in the t-path product signed graph is determined by the product of marks of the vertices in the signed graph $$ , 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 $$ which are switching equivalent to t-path product signed graphs $$ 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 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.