“School of Mathematics”
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| Paper IPM / M / 18021 |
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| Abstract: | |
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Let G be a graph, and let λ(G) denote the smallest eigenvalue
of G. First, we provide an upper bound for λ(G) based on induced bipartite subgraphs of G. Consequently, we extract two other upper bounds,
one relying on the average degrees of induced bipartite subgraphs and a
more explicit one in terms of the chromatic number and the independence
number of G. In particular, motivated by our bounds, we introduce two
graph invariants that are of interest on their own. Finally, special attention goes to the investigation of the sharpness of our bounds in various
classes of graphs as well as the comparison with an existing well-known
upper bound.
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