Combinatorial upper bounds for the smallest eigenvalue of a graph (Q6564137)

Let \(G\) be a graph, and let \(\lambda(G)\) denote the smallest eigenvalue of \(G\). In this paper, the authors first provide an upper bound for \(\lambda(G)\) based on induced bipartite subgraphs of \(G\). Consequently, the authors 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 their bounds, the authors introduce two graph invariants that were 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. This is a very interesting paper.