Wiener and Average Distance of Irregular Square-Cell Configuration

TL;DR

This paper extends the analysis of square-cell configurations from regular to irregular boundaries, deriving formulas for Wiener index and average distance to understand the impact of irregularity.

Contribution

It introduces generalized expressions for distances in irregular square-cell configurations, unifying symmetric and asymmetric cases.

Findings

Derived formulas for Wiener index of irregular configurations.

Analyzed how irregularity influences distance distribution.

Unified framework for symmetric and asymmetric square-cell graphs.

Abstract

A subgraph of the square lattice with all of its inner faces being 4-cycles is called a square-cell configuration. Prior work has provided explicit expressions for the total and average distances between vertex pairs in symmetric square-cell configurations, including well-structured families such as hexagonal square-cell configurations $H(n)$, trapezium square-cell configurations $T(n,k)$, and bitrapezium square-cell configurations $BT(n,k_1,k_2)$. In this article, we further extend the square-cell configuration from regular boundaries to irregular boundaries, which do not exhibit complete regularity or symmetry in their structure. We find the generalized expressions for the Wiener index and average distance of such irregular configurations, incorporating combinatorial and structural variations. Our results demonstrate how irregularity affects the growth and distribution of pairwise distances and provide a unifying framework that includes both symmetric and asymmetric square-cell graphs as exceptional cases. This generalization provides novel insights into the structural behaviour of square-cell frameworks characterized by complex or perturbed geometries.