Determinants of grids, tori, cylinders and M\"{o}bius ladders

TL;DR

This paper provides explicit formulas for determinants of various graph structures like grids, tori, cylinders, and Möbius ladders, simplifying previous recursive approaches with a unified technique.

Contribution

It introduces a concise proof method to derive explicit determinant formulas for multiple complex graph structures.

Findings

Explicit formulas for determinants of grids, tori, cylinders, and Möbius ladders.

Unified proof technique applicable to various graph determinants.

Simplification of previous recursive determinant calculations.

Abstract

Recently, Bie\~{n} [A. Bie\~{n}, The problem of singularity for planar grids, Discrete Math. 311 (2011), 921--931] obtained a recursive formula for the determinant of a grid. Also, recently, Pragel [D. Pragel, Determinants of box products of paths, Discrete Math. 312 (2012), 1844--1847], independently, obtained an explicit formula for this determinant. In this paper, we give a short proof for this problem. Furthermore, applying the same technique, we get explicit formulas for the determinant of a torus, a cylinder, and a M\"{o}bius ladder.