On Resistance Distance and Kirchhoff Index of Cacti Networks

Abstract

Resistance distance in electrical circuits measures how much a component or an entire circuit resists the flow of electric current. When dealing with intricate circuits, this term explicitly denotes the total resistance observed between any two points, which varies based on the configuration and resistance values of the components within the circuit. The Kirchhoff index is a metric used to quantify the mean resistance distance across all pairs of nodes in an electrical network. In graph theory, these networks are depicted as graphs with nodes representing electrical components and edges symbolizing the connecting wires. The resistance distance between any two nodes is calculated as if the graph were an electrical circuit, with each edge functioning as a resistor. We focus on a particular type of graph known as a cacti graph, denoted by \(\mathcal {C}(n,s)\), which features interconnected cycles that share a single common vertex, with n representing the total number of nodes and s the number of cycles. This paper explores cacti networks to establish the maximum possible values of the Kirchhoff index for these structures.