Exact two-point resistance, and the simple random walk on the complete graph minus $ N$ edges

TL;DR

This paper derives exact formulas for two-point resistance and random walk metrics on a modified complete graph, introducing Bejaia and Pisa numbers as new mathematical tools for analysis.

Contribution

It introduces Bejaia and Pisa numbers to generalize Fibonacci and Lucas numbers, enabling exact calculations of resistances and random walk times on a specific class of graphs.

Findings

Exact expressions for two-point resistance are obtained.

Formulas for first passage and mean first passage times are derived.

New mathematical sequences are introduced for graph analysis.

Abstract

An analytical approach is developed to obtain the exact expressions for the two-point resistance, and the total effective resistance of the complete graph minus $N$ edges of the opposite vertices. These expressions are written in terms of certain numbers that we introduced which we call the Bejaia and the Pisa numbers, these numbers are the natural generalizations of the bisected Fibonacci and Lucas numbers. The correspondence between random walks and the resistor networks is then used to obtain the exact expressions for the the first passage and mean first passage times on this graph.