19 research outputs found
Twice Q-Polynomial Distance-Regular Graphs
AbstractLetΓbe a distance-regular graph of diameter at least three. SupposeΓis Q-polynomial with respect to distinct eigenvaluesθandψofΓ. We calculate thekite numberf2j(x, y, z) from the intersection numbers ofΓand the dual eigenvalues toθandψ. This implies that the kite numbersfij(x, y, z) are independent of the verticesx,y,z. We use this result to show that the vertex neighborhood graphs ofΓare strongly regular and we calculate the parameters of these graphs from the kite numbersf2jand the intersection numbers ofΓ
The Terwilliger algebra of an almost-bipartite P- and Q-polynomial association scheme
Let denote a -class symmetric association scheme with , and
suppose is almost-bipartite P- and Q-polynomial. Let denote a vertex of
and let denote the corresponding Terwilliger algebra. We prove
that any irreducible -module is both thin and dual thin in the sense of
Terwilliger. We produce two bases for and describe the action of on
these bases. We prove that the isomorphism class of as a -module is
determined by two parameters, the dual endpoint and diameter of . We find a
recurrence which gives the multiplicities with which the irreducible
-modules occur in the standard module. We compute this multiplicity for
those irreducible -modules which have diameter at least .Comment: 22 page
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