8 research outputs found
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
Geometric aspects of 2-walk-regular graphs
A -walk-regular graph is a graph for which the number of walks of given
length between two vertices depends only on the distance between these two
vertices, as long as this distance is at most . Such graphs generalize
distance-regular graphs and -arc-transitive graphs. In this paper, we will
focus on 1- and in particular 2-walk-regular graphs, and study analogues of
certain results that are important for distance regular graphs. We will
generalize Delsarte's clique bound to 1-walk-regular graphs, Godsil's
multiplicity bound and Terwilliger's analysis of the local structure to
2-walk-regular graphs. We will show that 2-walk-regular graphs have a much
richer combinatorial structure than 1-walk-regular graphs, for example by
proving that there are finitely many non-geometric 2-walk-regular graphs with
given smallest eigenvalue and given diameter (a geometric graph is the point
graph of a special partial linear space); a result that is analogous to a
result on distance-regular graphs. Such a result does not hold for
1-walk-regular graphs, as our construction methods will show
Taut distance-regular graphs and the subconstituent algebra
We consider a bipartite distance-regular graph with diameter at least
4 and valency at least 3. We obtain upper and lower bounds for the local
eigenvalues of in terms of the intersection numbers of and the
eigenvalues of . Fix a vertex of and let denote the corresponding
subconstituent algebra. We give a detailed description of those thin
irreducible -modules that have endpoint 2 and dimension . In an earlier
paper the first author defined what it means for to be taut. We obtain
three characterizations of the taut condition, each of which involves the local
eigenvalues or the thin irreducible -modules mentioned above.Comment: 29 page
Krein parameters and antipodal tight graphs with diameter 3 and 4
AbstractWe determine which Krein parameters of nonbipartite antipodal distance-regular graphs of diameter 3 and 4 can vanish, and give combinatorial interpretations of their vanishing. We also study tight distance-regular graphs of diameter 3 and 4. In the case of diameter 3, tight graphs are precisely the Taylor graphs. In the case of antipodal distance-regular graphs of diameter 4, tight graphs are precisely the graphs for which the Krein parameter q114 vanishes
Remarks on pseudo-vertex-transitive graphs with small diameter
Let denote a -polynomial distance-regular graph with vertex set
and diameter . Let denote the adjacency matrix of . For a
vertex and for , let denote the projection
matrix to the th subconstituent space of with respect to . The
Terwilliger algebra of with respect to is the semisimple
subalgebra of generated by . Let denote a -vector space consisting
of complex column vectors with rows indexed by . We say is
pseudo-vertex-transitive whenever for any vertices , both (i) the
Terwilliger algebras and of are isomorphic; and (ii)
there exists a -vector space isomorphism such that
and for all . In this paper we discuss pseudo-vertex transitivity for
distance-regular graphs with diameter . In the case of diameter
two, a strongly regular graph is thin, and is
pseudo-vertex-transitive if and only if every local graph of has the
same spectrum. In the case of diameter three, Taylor graphs are thin and
pseudo-vertex-transitive. In the case of diameter four, antipodal tight graphs
are thin and pseudo-vertex-transitive.Comment: 29 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