Relations Between Resistance Distances of a Graph and its Complement or its Contraction

Yujun Yang

doi:10.5562/cca2318

Abstract

Abstract. The resistance distance between two vertices of a connected graph is defined as the net effective resistance between them when each edge of the graph is replaced by a resistor. In this paper, it is shown that the product of resistance distances between any pair of vertices in a simple graph and in its connected complement is less than or equal to 3. Meanwhile, a relation between resistance distances of a graph and its contraction is obtained in a special case. (doi: 10.5562/cca2318) Keywords: resistance distance, graph complement, graph contraction, Rayleigh's short-cut principle INTRODUCTION Two decades ago, a novel distance function on graphs was identified by Klein and Randic. 1 They viewed a connected graph as an electrical network by imagining that fixed resistors are assigned to each edge. Then they proved that the effective resistance between pairs of vertices is a distance function on the graph and named this new distance function resistance distance . As a central component of electric circuit theory, effective resistance have long been studied in physics and engineering, dating back to Kirchhoff

Highlights

Two decades ago, a novel distance function on graphs was identified by Klein and Randić.[1]
They viewed a connected graph as an electrical network by imagining that fixed resistors are assigned to each edge
They proved that the effective resistance between pairs of vertices is a distance function on the graph and named this new distance function resistance distance

Summary

Introduction

A novel distance function on graphs was identified by Klein and Randić.[1]. RELATIONS BETWEEN RESISTANCE DISTANCES OF A GRAPH AND ITS COMPLEMENT We consider resistance distance of a graph and its complement. We need the formula for computing resistance distances of circulant graphs.

Results

Conclusion