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

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. Finally, edge clique partitions of additional infinite families of connected graphs and their relations with the least eigenvalues are presented.