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

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 G₁ and G₂ be two vertex disjoint graphs. The central vertex join of G₁ and G₂ is the graph G₁̇ G₂, is obtained from C(G₁) and G₂ by joining each vertex of G₁ with every vertex of G₂. The central edge join of G₁ and G₂ is the graph G₁ G₂, is obtained from C(G₁) and G₂ by joining each vertex corresponding to the edges of G₁ with every vertex of G₂. In this article, we obtain formulae for the resistance distance and Kirchhoff index of G₁̇ G₂ and G₁ G₂. In addition, we provide the resistance distance, Kirchhoff index, and Kemeny's constant of the central graph of a graph.