Sharp bounds on the least eigenvalue of a graph determined from edge clique partitions

TL;DR

This paper establishes precise bounds on the smallest eigenvalue of any graph using edge clique partitions, providing conditions for these bounds to be tight and applying results to the n-Queens graph.

Contribution

It introduces sharp bounds on the least eigenvalue based on edge clique partitions and characterizes when these bounds are attained, with applications to specific graphs.

Findings

Least eigenvalue of the n-Queens graph is -4 for all n ≥ 4

Multiplicity of the least eigenvalue in the n-Queens graph is (n-3)^2

Derived new results on edge clique partition graph parameters

Abstract

Sharp bounds on the least eigenvalue of an arbitrary graph are presented. Necessary and sufficient (just sufficient) conditions for the lower (upper) bound to be attained are deduced using edge clique partitions. As an application, we prove that the least eigenvalue of the $n$-Queens' graph $\mathcal{Q}(n)$ is equal to $-4$ for every $n \ge 4$ and it is also proven that the multiplicity of this eigenvalue is $(n-3)^2$. Additionally, some results on the edge clique partition graph parameters are obtained.