On Distance-Regular Graphs with Smallest Eigenvalue at Least −m-m

A non-complete geometric distance-regular graph is the point graph of a partial geometry in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for fixed integer $m\geq 2$, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least $-m$, diameter at least three and intersection number $c_2 \geq 2$

An inequality involving the second largest and smallest eigenvalue of a distance-regular graph

For a distance-regular graph with second largest eigenvalue (resp. smallest eigenvalue) \mu1 (resp. \muD) we show that (\mu1+1)(\muD+1)<= -b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.Comment: 15 pages, this is submitted to Linear Algebra and Applications

Shilla distance-regular graphs

A Shilla distance-regular graph G (say with valency k) is a distance-regular graph with diameter 3 such that its second largest eigenvalue equals to a3. We will show that a3 divides k for a Shilla distance-regular graph G, and for G we define b=b(G):=k/a3. In this paper we will show that there are finitely many Shilla distance-regular graphs G with fixed b(G)>=2. Also, we will classify Shilla distance-regular graphs with b(G)=2 and b(G)=3. Furthermore, we will give a new existence condition for distance-regular graphs, in general.Comment: 14 page

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page