Resistance distance and sharp bounds of two-mode electrical networks

Abstract

Electrical networks are ubiquitous in our daily lives, ranging from small integrated circuits to large-scale power systems. These networks can be easily represented as graphs, where edges represent connections and vertices represent electric nodes. The concept of resistance distance originates from electrical networks, with this term used because of its physical interpretation, where every edge in a graph G is assumed to have a unit resistor. The applications of resistance distance extend to various fields such as electrical engineering, physics, and computer science. It is particularly useful in investigating the flow of electrical current in a network and determining the shortest path between two vertices. In this work, we have investigated seven different resistance distance-based indices of bipartite networks and derived general formulae for them; the sharp bounds with respect to these resistance distance indices are also identified. Additionally, we introduced a novel resistance distance topological index, the Multiplicative Eccentric Resistance Harary Index, and derived general formula for it. The sharp bounds with respect to this newly introduced index are also identified for bipartite networks.