Resistance distance and Kirchhoff index in central vertex join and central edge join of two graphs

Haritha T; Chithra A

paper

Resistance distance and Kirchhoff index in central vertex join and central edge join of two graphs

arXiv:2404.06317

Abstract

The central graph $C (G)$ of a graph $G$ is the graph obtained by inserting a new vertex into each edge of $G$ exactly once and joining all the non-adjacent vertices in $G$ . Let $G_{1}$ and $G_{2}$ be two vertex disjoint graphs. The central vertex join of $G_{1}$ and $G_{2}$ is the graph $G_{1} \lor ˙ G_{2}$ , is obtained from $C (G_{1})$ and $G_{2}$ by joining each vertex of $G_{1}$ with every vertex of $G_{2}$ . The central edge join of $G_{1}$ and $G_{2}$ is the graph $G_{1} ⊻ G_{2}$ , is obtained from $C (G_{1})$ and $G_{2}$ by joining each vertex corresponding to the edges of $G_{1}$ with every vertex of $G_{2}$ . In this article, we obtain formulae for the resistance distance and Kirchhoff index of $G_{1} \lor ˙ G_{2}$ and $G_{1} ⊻ G_{2}$ . In addition, we provide the resistance distance, Kirchhoff index, and Kemeny's constant of the central graph of a graph.