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

Domingos M. Cardoso, Inês Serôdio Costa, Rui Duarte

Published 2022-01-04, updated 2022-02-15Version 3

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.