173 research outputs found
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
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
Distance-regular graph with large a1 or c2
In this paper, we study distance-regular graphs 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 . 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
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