Abstract
AbstractSharp 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 $${\mathcal {Q}}(n)$$
Q
(
n
)
is equal to $$-4$$
-
4
for every $$n \ge 4$$
n
≥
4
and it is also proven that the multiplicity of this eigenvalue is $$(n-3)^2$$
(
n
-
3
)
2
. Finally, edge clique partitions of additional infinite families of connected graphs and their relations with the least eigenvalues are presented.
Funder
Fundação para a Ciência e a Tecnologia
Centro de Investigação e Desenvolvimento em Matemática e Aplicações
Publisher
Springer Science and Business Media LLC
Subject
Discrete Mathematics and Combinatorics,Algebra and Number Theory
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