On distance-regular graphs with smallest eigenvalue at least −m

Abstract

A non-complete geometric distance-regular graph is the point graph of a partial linear space in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for a fixed integer m 2, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least −m, diameter at least three and intersection number c 2 2.

There Does Not Exist a Distance-Regular Graph with Intersection Array $$\{80, 54,12; 1, 6, 60\}$$

“…[5,Lemma 4.1]) Suppose that Γ is a geometric distanceregular graph with valency k ≥ 2, diameter D ≥ 2 and smallest eigenvalue θ D . Then the following hold: [5,Lemma 4.2]) Let Γ be a geometric distance-regular graph with diameter D ≥ 2 and intersection number c 2 ≥ 2. Then τ 2 ≥ ψ 1 holds.…”

Distance-regular graphs without 4-claws

“…Since the smallest eigenvalue θ D of Γ is bounded below (see Lemma 2.7), it follows by [27] that there are only finitely many non-geometric distance-regular graphs. In particular, they showed that the valency k is bounded above as O(m 2 1 ), where m 1 is the multiplicity of second largest eigenvalue.…”

“…In particular, Spielman [35] improved the complexity of isomorphism testing of strongly regular graphs, found by Babai [1], using the following result by Neumaier [32]: for a fixed integer θ ≥ 2, there are only finitely many coconnected non-geometric distance-regular graphs with smallest eigenvalue at least −θ and diameter 2. Koolen and Bang [27] generalised this as follows: there are only finitely many coconnected non-geometric distance-regular graphs with smallest eigenvalue at least −θ and with given diameter D ≥ 2 or having the intersection number c 2 ≥ 2 (in fact, the Bannai-Ito conjecture recently proved by Bang et al [5] shows that the latter condition on c 2 is not necessary).…”

“…This shows that the above-mentioned result of Neumaier is the best we can hope for the case of distance-regular graphs of diameter two. The situation for distance-regular graphs with larger diameter seems to be different: Koolen and Bang [27] conjectured that, for a fixed integer θ ≥ 2, any geometric distance-regular graph with smallest eigenvalue −θ, diameter D ≥ 3 and c 2 ≥ 2 is either a Johnson graph, a Grassmann graph, a Hamming graph, a bilinear forms graph or the number of vertices is bounded above by a function of θ.…”

“…Koolen and Bang [27] showed that Γ is geometric, if a 1 ≥ ⌊θ D ⌋ 2 c 2 holds. Moreover, they [19,Proposition 9.7] showed that a distance-regular graph Γ of diameter D ≥ 2 having valency k ≥ max{t 2 − t, Clearly, a connected graph without 2-claws is a complete graph.…”