12 research outputs found
The twisted Grassmann graph is the block graph of a design
In this note, we show that the twisted Grassmann graph constructed by van Dam
and Koolen is the block graph of the design constructed by Jungnickel and
Tonchev. We also show that the full automorphism group of the design is
isomorphic to the full automorphism group of the twisted Grassmann graph.Comment: 5 pages. A section on the automorphism group has been adde
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
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