As one of the serial papers on suborbits of point stabilizers in classical groups on the last subconstituent of dual polar graphs, the corresponding problem for orthogonal dual polar graphs over a ...finite field of odd characteristic is discussed in this paper. We determine all the suborbits of a point-stabilizer in the orthogonal group on the last subconstituent, and calculate the length of each suborbit. Moreover, we discuss the quasi-strongly regular graphs and the association schemes based on the last subconstituent, respectively.
The power graph PG of a finite group G is the graph with the vertex set G, where two distinct vertices are adjacent if one is a power of the other. We first show that PG has a transitive orientation, ...so it is a perfect graph and its core is a complete graph. Then we use the poset on all cyclic subgroups of G (under usual inclusion) to characterize the structure of PG. Finally, a closed formula for the metric dimension of PG is established. As an application, we compute the metric dimension of the power graph of a cyclic group.
Suppose
F
q
n
+
l
denotes the
(
n
+
l
)
-dimensional vector space over a finite field
F
q
and
GL
n
+
l
,
n
(
F
q
)
denotes the corresponding singular general linear group. All the subspaces of type
(
...m
,
k
)
form an orbit under
GL
n
+
l
,
n
(
F
q
)
, denoted by
M
(
m
,
k
;
n
+
l
,
n
)
. Let
Λ
be the set of all the orbitals of
(
GL
n
+
l
,
n
(
F
q
)
,
M
(
m
,
k
;
n
+
l
,
n
)
)
. Then
(
M
(
m
,
k
;
n
+
l
,
n
)
,
Λ
)
is a symmetric association scheme. In this paper, we determine all the orbitals and the rank of
(
GL
n
+
l
,
n
(
F
q
)
,
M
(
m
,
k
;
n
+
l
,
n
)
)
, calculate the length of each suborbit. Finally, we compute all the intersection numbers of the symmetric association scheme
(
M
(
m
,
k
;
n
+
l
,
n
)
,
Λ
)
, where
k
=
1
or
k
=
l
-
1
.
This paper provides some new families of symmetric association schemes based on maximal totally isotropic subspaces in (singular) pseudo-symplectic spaces. All intersection numbers of these schemes ...are computed.
Let
Γ
=
(
X
,
R
)
denote a
d
-bounded distance-regular graph with diameter
d
≥
3
. A regular strongly closed subgraph of
Γ
is said to be a subspace of
Γ
. For
x
∈
X
, let
P
(
x
)
be the set of all ...subspaces of
Γ
containing
x
. For each
i
=
1
,
2
,
…
,
d
−
1
, let
Δ
0
be a fixed subspace with diameter
d
−
i
in
P
(
x
)
, and let
ℒ
(
d
,
i
)
=
{
Δ
∈
P
(
x
)
∣
Δ
+
Δ
0
=
Γ
,
d
(
Δ
)
=
d
(
Δ
∩
Δ
0
)
+
i
}
∪
{
0̸
}
.
If we define the partial order on
ℒ
(
d
,
i
)
by ordinary inclusion (resp. reverse inclusion), then
ℒ
(
d
,
i
)
is a finite poset, denoted by
ℒ
O
(
d
,
i
)
(resp.
ℒ
R
(
d
,
i
)
). In the present paper we show that both
ℒ
O
(
d
,
i
)
and
ℒ
R
(
d
,
i
)
are atomic, and compute their characteristic polynomials.
The generalized k-connectivity κk(G) of a graph G, which was introduced by Chartrand et al. (1984) is a generalization of the concept of vertex connectivity. Let G and H be nontrivial connected ...graphs. Recently, Li et al. (2012) gave a lower bound for the generalized 3-connectivity of the Cartesian product graph G□H and proposed a conjecture for the case that H is 3-connected. In this paper, we give two different forms of lower bounds for the generalized 3-connectivity of Cartesian product graphs. The first lower bound is stronger than theirs, and the second confirms their conjecture.
•We obtain the upper bounds for the number of edges in even-cycle-free subgraphs of the complete transposition graphs, which are a family of bipartite and arctransitive graphs.•A family of new ...auxiliary graphs related to the complete transposition graphs are constructed and used in order to obtain the upper bounds.•A new Ramsey-type result for the complete transposition graphs is obtained.
Given graphs G and H, the generalized Turán number ex(G,H) is the maximum number of edges in an H-free subgraph of G. In this paper, we obtain an asymptotic upper bound on ex(CTn,C2ℓ) for any n≥3 and ℓ≥2, where C2ℓ is the cycle of length 2ℓ and CTn is the complete transposition graph which is defined as the Cayley graph on the symmetric group Sn with respect to the set of all transpositions of Sn.
The paper provides the construction of association schemes on the sets of the maximal totally isotropic subspaces in singular classical spaces. All intersection numbers of these schemes are computed.