Let
G be a distance-regular graph. Let
A denote the adjacency matrix of
G. Fix a vertex
x of
G. For each
i
(0⩽i⩽D)
, let
E
i
∗=E
i
∗(x)
denote the projection onto the
ith subconstituent of
G with ...respect to
x. Let
T(x) denote the
C
-algebra generated by
A and
{E
i
∗
|
0⩽i⩽D}
. We call
T(x) the
Terwilliger algebra of G with respect to x. An irreducible
T(x)-module
W is said to be
thin if dim
E
i
∗W⩽1
for
0⩽i⩽D. The graph
G is
thin if for each vertex
x of
G, every irreducible
T(x)-module is thin. A distance-regular graph
G=(
X,
R)
with diameter
D
is said to be
almost-bipartite if the intersection numbers satisfy
a
i(
G)=0
(0⩽i⩽
D−1)
and
a
D
(
G)≠0
. Let
G=(
X,
R)
be an almost-bipartite distance-regular graph with diameter
D
. Then there is a distance-regular graph
G=(X,R) of diameter
D=2
D+1
which is an antipodal 2-cover, and a 2-to-1 surjection
π
:
X→
X
which preserves adjacency. It is known that
G
and
G determine each other, up to isomorphism. We investigate the relationship between the Terwilliger algebras and their module structures of two graphs related in this way. In particular, we show that
G
is thin if and only if
G is thin.
Given a vertex subset C of a distance-regular graph $\Gamma$ on n vertices, it is shown that C is a completely regular code if and only if the number of vertices at maximum distance from C satisfies ...an expression in terms of the spectrum of $\Gamma$ and some mean numbers computed from the distances among the vertices of C (the so-called "inner distribution" of C). For such codes, this result can be seen as an improvement of Delsarte's linear programming method, since it gives stronger necessary conditions for their existence. As an application, a purely spectral characterization of those distance-regular graphs which are "edge-distance-regular" (that is, with every edge being a completely regular code with the same parameters) is derived.
Let
F
q
be the finite field with
q elements. Denote by
Γ
(
δ)
the dual polar graph of (2
ν+
δ)-dimensional orthogonal space over
F
q
, where
δ=0, 1 or 2. For any vertex
P of
Γ
(
δ)
, all ...subconstituents
Γ
i
(δ)(P)
(1⩽i⩽ν)
of
Γ
(
δ)
are studied, and it is proved that
Γ
i
(
δ)
(
P) is isomorphic to
ν
i
q·Λ
(i,δ),
where
Λ
(
i,
δ)
is a subgraph of the graph of
i×(
i+
δ) matrices over
F
q
. Moreover, some properties of the graph
Λ
(
i,
δ)
are also studied. In particular, it is shown that
Λ
(
i,
δ)
is edge-regular. Furthermore, both
Λ
(2,1) and
Λ
(3,1) are distance-regular with intersection arrays
{q
2−1,q
2−q,1;1,q,q
2−1}
and
{q
3−1,q
3−q,q
3−q
2+1;1,q,q
2−1},
respectively.
A generalization of strong regularity around a vertex subset C of a graph γ , which makes sense even if γ is non-regular, is studied. Such a structure appears, together with a kind of ...distance-regularity around C , when an spectral bound concerning the so-called predistance polynomial of C is attained. As a main consequence of these results, it is shown that a regular (connected) graph γ with d + 1 distinct eigenvalues is distance-regular, and its distance- d graph γ d is strongly regular with parameters a = c , if and only if the number of vertices at distance d from each vertex satisfies an expression which depends only on the order of γ and the different eigenvalues of γ .
A spin model is a square matrixWsatisfying certain conditions which ensure that it yields an invariant of knots and links via a statistical mechanical construction of V. F. R. Jones. Recently F. ...Jaeger gave a topological construction for each spin modelWof an association scheme which containsWin its Bose–Mesner algebra. Shortly thereafter, K. Nomura gave a simple algebraic construction of such a Bose–Mesner algebraN(W). In this paper we study the caseW∈A⊆N(W), where A is the Bose–Mesner algebra of a distance-regular graph. We show the following results. LetΓ=(X,R) be a distance-regular graph of diameterd>1 such that the Bose–Mesner algebra A ofΓsatisfiesW∈A⊆N(W) for some spin modelWonX. WriteW=∑di=0tiAi, whereAidenotes theith adjacency matrix. Setxi=t−1i−1tiandp=x−11x2. Thenxi=pi−1x1holds for alli. Moreover, the eigenvalues and the intersection numbers ofΓare rational functions ofx1andp.
Let $G$ be a regular (connected) graph with $n$ vertices and $d+1$ distinct eigenvalues. As a main result, it is shown that $G$ is an $r$-antipodal distance-regular graph if and only if the distance ...graph $G_d$ is constituted by disjoint copies of the complete graph $K_r$, with $r$ satisfying an expression in terms of $n$ and the distinct eigenvalues.
Cameron-Liebler sets were originally defined as collections of lines (“line classes”) in PG(3,q) sharing certain properties with line classes of symmetric tactical decompositions. While there are ...many equivalent characterisations, these objects are defined as sets of lines whose characteristic vector lies in the image of the transpose of the point-line incidence matrix of PG(3,q), and so combinatorially they behave like a union of pairwise disjoint point-pencils. Recently, the concept of a Cameron-Liebler set has been generalised to several other settings. In this article we introduce Cameron-Liebler sets of generators in finite classical polar spaces. For each of the polar spaces we give a list of characterisations that mirrors those for Cameron-Liebler line sets, and also prove some classification results.
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