Resistance distance and Kirchhoff index for a class of graphs (Q1720348)

Summary: Let \(G [F, V_k, H_v]\) be the graph with \(k\) pockets, where \(F\) is a simple graph of order \(n \geq 1\), \(V_k = \{v_1, v_2, \ldots, v_k \}\) is a subset of the vertex set of \(F, H_v\) is a simple graph of order \(m \geq 2\), and \(v\) is a specified vertex of \(H_v\). Also let \(G [F, E_k, H_{u v}]\) be the graph with \(k\) edge pockets, where \(F\) is a simple graph of order \(n \geq 2, E_k = \{e_1, e_2, \ldots e_k \}\) is a subset of the edge set of \(F, H_{u v}\) is a simple graph of order \(m \geq 3\), and \(u v\) is a specified edge of \(H_{u v}\) such that \(H_{u v} - u\) is isomorphic to \(H_{u v} - v\). In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of \(G [F, V_k, H_v]\) and \(G [F, E_k, H_{u v}]\) in terms of the resistance distance and Kirchhoff index \(F, H_v\) and \(F, H_{u v}\), respectively.