87 research outputs found

### 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

### 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 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

### 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

- β¦