Geodesic
Some Observations on the Smallest Adjacency Eigenvalue of a Graph
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 467-493

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In this paper, we discuss various connections between the smallest eigenvalue of the adjacency matrix of a graph and its structure. There are several techniques for obtaining upper bounds on the smallest eigenvalue, and some of them are based on Rayleigh quotients, Cauchy interlacing using induced subgraphs, and Haemers interlacing with vertex partitions and quotient matrices. In this paper, we are interested in obtaining lower bounds for the smallest eigenvalue. Motivated by results on line graphs and generalized line graphs, we show how graph decompositions can be used to obtain such lower bounds.
Keywords: graph spectrum, smallest eigenvalue, adjacency matrix, graph decomposition, clique partition, claw-free graphs, maximum cut 
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Cioabă, Sebastian M.; Elzinga, Randall J.; Gregory, David A. Some Observations on the Smallest Adjacency Eigenvalue of a Graph. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 467-493. http://geodesic.mathdoc.fr/item/DMGT_2020_40_2_a7/