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

Abstract

A signed graph $\Sigma$ is a pair $\Sigma =\left({\Sigma }^u,\sigma \right)$that consists of a graph $\left({\Sigma }^u,E\right)$ and a sign mapping called signature $\sigma$ from E to the sign group $\left\{+,-\right\}$. In this paper, we discuss the t-path product signed graph ${\widehat{\left(\Sigma \right)}}_t$where vertex set of ${\widehat{\left(\Sigma \right)}}_t$ is the same as that of $\Sigma$ and two vertices are adjacent if there is a path of length t, between them in the signed graph $\Sigma$. 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 $\Sigma$, 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 $\Sigma$ which are switching equivalent to t-path product signed graphs ${\widehat{\left(\Sigma \right)}}_t$ for $t=2,3$ which are switching equivalent to $\Sigma$ and also the negation of the signed graph ŋ$\left(\Sigma \right)$ that are switching equivalent to ${\widehat{\left(\Sigma \right)}}_t$ for $t=2,3$. We also characterize signed graphs that are switching equivalent to $t$-distance signed graph ${\left(\overline{\Sigma }\right)}_t$ for $t=2$ where 2-distance signed graph ${\left(\overline{\Sigma }\right)}_2=\left(V^{\prime },E^{\prime },{\sigma }^{\prime }\right)$ defined as follows: the vertex set is same as the original signed graph $\Sigma$ and two vertices $u,v\in {\left(\overline{\Sigma }\right)}_2$, are adjacent if and only if there exists a distance of length two in $\Sigma$. The edge $uv\in {\left(\overline{\Sigma }\right)}_2$ is negative if and only if all the edges, in all the distances of length two in $\Sigma$ 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.