Wiener and Average Distance of Irregular Square-Cell Configuration

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.