On the determinant of hexagonal grids $H_{k,n}$

TL;DR

This paper derives an explicit formula for the determinant of hexagonal grid graphs $H_{k,n}$, showing they are all non-singular, by developing methods to transform weighted graphs without changing their adjacency matrix determinants.

Contribution

Introduces novel methods for transforming weighted graphs that preserve determinants, enabling the calculation of determinants for all hexagonal grid graphs $H_{k,n}$.

Findings

All graphs $H_{k,n}$ are non-singular.

Explicit formula for the determinant of $H_{k,n}$.

Methods for determinant-preserving graph transformations.

Abstract

We analyse the problem of singularity of graphs for hexagonal grid graphs. We introduce methods for transforming weighted graph, which do not change the determinant of adjacency matrix. We use these methods to calculate the determinant of all hexagonal grid graphs which describe certain hexagon-shaped benzenoid systems. The final result is the explicit formula for the determinant of graphs $H_{k,n}$. From the theorem we draw the conclusion, that all graphs of this kind are non-singular.