The resistance distance \(r_{G}(u,v)\) between two vertices u and v of a graph G is defined as the net effective resistance between them in the electric network constructed from G by replacing each edge with a unit resistor. The Kirchhoff index Kf(G) is defined as the sum of resistance distances between all pairs of vertices. Let \(L^{m}_{n}\) be a \(K_n\) -chain network with m complete graphs. Then identifying the opposite lateral edges of \(L^{m}_{n}\) in an order way yields the \(K_n\) -ring, denoted by \(C^{m}_{n}\) . In this paper, we first construct a new equivalent network transformation on complete graphs. Then utilize combinatorial and electrical network approaches, we give explicit formula for the resistance distances between any two vertices in \(L^{m}_{n}\) and \(C^{m}_{n}\) . Further, the closed-form formulas of the Kirchhoff index for \(L^{m}_{n}\) and \(C^{m}_{n}\) are also obtained. In addition, our results contain the main results of [Symmetry. 15(5) (2023) 1122] and [Phys. Scr. 98(4) (2023) 045222] as special cases.
