A remark on the intersection arrays of distance-regular graphs

AbstractIt is shown that the number of columns of type (1, 1, k − 2) in the intersection array of a distance-regular graph with valency k and girth >3 is at most four

The distance-regular graphs such that all of its second largest local eigenvalues are at most one

In this paper, we classify distance regular graphs such that all of its second largest local eigenvalues are at most one. Also we discuss the consequences for the smallest eigenvalue of a distance-regular graph. These extend a result by the first author, who classified the distance-regular graph with smallest eigenvalue $-1-\frac{b_1}{2}$.Comment: 16 pages, this is submitted to Linear Algebra and Application

Distance-regular graph with large a1 or c2

In this paper, we study distance-regular graphs $\Gamma$ that have a pair of distinct vertices, say x and y, such that the number of common neighbors of x and y is about half the valency of $\Gamma$. We show that if the diameter is at least three, then such a graph, besides a finite number of exceptions, is a Taylor graph, bipartite with diameter three or a line graph.Comment: We submited this manuscript to JCT

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

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 Cayley graphs with small valency

We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most $4$, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency $5$, and the Cayley graphs among all distance-regular graphs with girth $3$ and valency $6$ or $7$. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some ``exceptional'' distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs.Comment: 19 pages, 4 table

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