Computation of resistance distances and Kirchhoff indices for two classes of graphs

For any two vertices u and v of a connected graph G, the resistance distance between u and v is defined as the effective resistance between them in the corresponding electrical network created by placing a unit resistor on each edge of G. The Kirchhoff index of G is defined as the sum of resistance distances between all pairs of vertices in G. Let $K_r^{-}$ be the graph obtained from the complete graph $K_r$ by deleting an edge. In this paper, we consider two classes of graphs formed by $K_r^{-}$, namely the string graph of $K_r^{-}$ and the ring graph of $K_r^{-}$, which are denoted by $S(K_r^{-},n)$ and $R(K_r^{-},n)$, respectively. By using combinatorial and electrical network approaches, we obtain the formulae for resistance distances and Kirchhoff indices of $S(K_r^{-},n)$ and $R(K_r^{-},n)$, which generalizes the results by Sardar et al. (2024) [25].