A family of regular integral graphs and its application to the $n$-Queens' graph

TL;DR

This paper studies a family of regular integral graphs called triangular graphs, analyzing their spectra and eigenvectors, and applies these findings to decompose the complex $n$-Queens' graph into simpler components, providing bounds on its eigenvalues.

Contribution

It introduces and analyzes the spectral properties of the triangular graphs and applies these insights to decompose the $n$-Queens' graph and estimate its eigenvalues.

Findings

Spectral structure of triangular graphs is characterized.

Decomposition of $n$-Queens' graph into triangular, clique, and bipartite components.

Bounds on eigenvalues of the $n$-Queens' graph are established.

Abstract

A family of regular integral graphs introduced in [I.F.S. Costa, The $n$-Queens graph and its generalizations, Ph.D. Thesis, University of Aveiro 2024], denoted by ${\cal T}(n)$ and herein called triangular graphs, is analysed. In this analysis, the consistent structure of the graph spectra and the patterns of the corresponding eigenvectors are highlighted. The properties of these graphs are examined and applied to the decomposition of the $n$-Queens' graph into three distinct families: a family of a single graph whose components are two triangular graphs, ${\cal T}(n)$ and ${\cal T}(n-1)$, a family of a single graph whose components are cliques and a family of complete bipartite graphs. Finally, using Weyl's inequalities, we introduce some techniques to establish lower and upper bounds on the eigenvalues of the $n$-Queens' graph.