Let Γ denote a Q-polynomial distance-regular graph with vertex set X and diameter D. Let A denote the adjacency matrix of Γ. For a vertex x∈X and for 0≤i≤D, let Ei⁎(x) denote the projection matrix to ...the ith subconstituent space of Γ with respect to x. The Terwilliger algebra T(x) of Γ with respect to x is the semisimple subalgebra of MatX(C) generated by A,E0⁎(x),E1⁎(x),…,ED⁎(x). Let V denote a C-vector space consisting of complex column vectors with rows indexed by X. We say Γ is pseudo-vertex-transitive whenever for any vertices x,y∈X, there exists a C-vector space isomorphism ρ:V→V such that (ρA−Aρ)V=0 and (ρEi⁎(x)−Ei⁎(y)ρ)V=0 for all 0≤i≤D. In this paper, we discuss pseudo-vertex transitivity for distance-regular graphs with diameter D∈{2,3,4}. For D=2, we show that a strongly regular graph is pseudo-vertex-transitive if and only if all its local graphs have the same spectrum. For D=3, we consider the Taylor graphs and show that they are pseudo-vertex transitive. For D=4, we consider the antipodal tight graphs and show that they are pseudo-vertex transitive.
The Jevons group
is an isometry group of the Hamming metric on the
-dimensional vector space
over
(2). It is generated by the group of all permutation (
×
)-matrices over
(2) and the translation ...group on
. Earlier the authors of the present paper classified the submetrics of the Hamming metric on
for
⩾ 4, and all overgroups of
which are isometry groups of these overmetrics. In turn, each overgroup of
is known to define orbital graphs whose “natural” metrics are submetrics of the Hamming metric. The authors also described all distance-transitive orbital graphs of overgroups of the Jevons group
. In the present paper we classify the distance-transitive orbital graphs of overgroups of the Jevons group. In particular, we show that some distance-transitive orbital graphs are isomorphic to the following classes: the complete graph
, the complete bipartite graph
, the halved (
+ 1)-cube, the folded (
+ 1)-cube, the graphs of alternating forms, the Taylor graph, the Hadamard graph, and incidence graphs of square designs.
A
resolving set
for a graph
Γ
is a collection of vertices
S
, chosen so that for each vertex
v
, the list of distances from
v
to the members of
S
uniquely specifies
v
. The
metric dimension
of
Γ
is ...the smallest size of a resolving set for
Γ
. Much attention has been paid to the metric dimension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimension of primitive distance-regular graphs in terms of their parameters. We show how the metric dimension of an imprimitive distance-regular graph can be related to that of its halved and folded graphs. We also consider infinite families (including Taylor graphs and the incidence graphs of certain symmetric designs) where more precise results are possible.
In this paper, we study the distance-regular graphs
Γ 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
Γ. 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.
We 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
q
11
4 vanishes.
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