On the order of regular graphs with fixed second largest eigenvalue

Problems on Graphs with Fixed Smallest Eigenvalue

“…Note that the smallest eigenvalue of K 2t goes to −∞ as t goes to ∞. (For a proof, see [95,Lemma 3.2].) It is fairly easy to see that the smallest eigenvalue of the t-claw K 1,t also goes to −∞ as t goes to ∞, as λ min (K 1,t ) = − √ t.…”

“…For the definition of a thin Q-polynomial distance-regular graph, see Appendix A. Now we present an important tool shown by Yang and Koolen [95], by using Hoffman graphs. Next we give an upper bound on the parameter c for co-edge regular graphs in terms of its smallest eigenvalue, due to Yang and Koolen [95].…”

“…Now we present an important tool shown by Yang and Koolen [95], by using Hoffman graphs. Next we give an upper bound on the parameter c for co-edge regular graphs in terms of its smallest eigenvalue, due to Yang and Koolen [95]. They showed the following: Their proof used Proposition 6.3 and the Alon-Boppana Theorem.…”

“…To prove Theorem 1.1 and Theorem 1.2, Neumaier used the Krein parameters of the underlying association scheme. In [7], Yang and Koolen showed the following result on co-edge regular graphs.…”

A spectral characterization of the $s$-clique extension of the triangular graphs

“…The reason for the valency condition is, that in [12], it was shown that for λ ≥ 2, there exist constants C(λ) such that a connected k-regular co-edge-regular graph with order v and smallest eigenvalue at least −λ satisfies one of the following:…”