# The resistance distance and Kirchhoff index on quadrilateral graph and pentagonal graph

## TL;DR

This paper derives explicit formulas for resistance distance and Kirchhoff index in quadrilateral and pentagonal graphs, which are constructed by replacing edges in an arbitrary graph with specific parallel paths.

## Contribution

It provides the first closed-form formulas for resistance distance and Kirchhoff index on these specific graph transformations.

## Key findings

- Closed-form formulas for resistance distance on Q(G) and W(G).
- Closed-form formulas for Kirchhoff index on Q(G) and W(G).
- Applicable to arbitrary base graphs G.

## Abstract

The quadrilateral graph Q(G) is obtained from G by replacing each edge in G with two parallel paths of length 1 and 3, whereas the pentagonal graph W(G) is obtained from G by replacing each edge in G with two parallel paths of length 1 and 4. In this paper, closed-form formulas of resistance distance and Kirchhoff index for quadrilateral graph and pentagonal graph are obtained whenever G is an arbitrary graph.