The Smith and Critical Groups of the Square Rook's Graph and its Complement

TL;DR

This paper computes the Smith and critical groups of the square rook's graph and its complement, confirming a 1986 conjecture by analyzing their adjacency and Laplacian matrices.

Contribution

It provides the first explicit determination of the Smith and critical groups for these graphs, verifying a longstanding conjecture.

Findings

Smith and critical groups of $R_n$ and its complement are explicitly computed.

The Smith normal form of adjacency and Laplacian matrices are determined.

A 1986 conjecture of Rushanan is verified.

Abstract

Let $R_{n}$ denote the graph with vertex set consisting of the squares of an $n \times n$ grid, with two squares of the grid adjacent when they lie in the same row or column. This is the square rook's graph, and can also be thought of as the Cartesian product of two complete graphs of order $n$, or the line graph of the complete bipartite graph $K_{n,n}$. In this paper we compute the Smith group and critical group of the graph $R_{n}$ and its complement. This is equivalent to determining the Smith normal form of both the adjacency and Laplacian matrix of each of these graphs. In doing so we verify a 1986 conjecture of Rushanan.