A weakly distance-regular digraph is thick if its attached scheme is regular. In this paper, we show that each commutative thick weakly distance-regular digraph has a thick weakly distance-regular ...subdigraph such that the corresponding quotient digraph falls into six families of thick weakly distance-regular digraphs up to isomorphism.
Weakly distance-regular digraphs are a natural directed version of distance-regular graphs. In Wang and Suzuki (Discrete Math 264:225–236, 2003), the third author and Suzuki proposed a question when ...an orientation of a distance-regular graph defines a weakly distance-regular digraph. In this paper, we initiate this project and classify all commutative weakly distance-regular digraphs whose underlying graphs are Hamming graphs, folded
n
-cubes and Doob graphs, respectively.
The power graph
Γ
G
of a finite group
G
is the graph with the vertex set
G
, where two distinct elements are adjacent if and only if one is a power of the other. An
L
(2, 1)-labeling of a graph
Γ
is ...an assignment of labels from nonnegative integers to all vertices of
Γ
such that vertices at distance two get different labels and adjacent vertices get labels that are at least 2 apart. The lambda number of
Γ
, denoted by
λ
(
Γ
)
, is the minimum span or range over all
L
(2, 1)-labelings of
Γ
. In this paper, we obtain bounds for
λ
(
Γ
G
)
and give necessary and sufficient conditions when the bounds are attained. As applications, we compute the exact value of
λ
(
Γ
G
)
if
G
is a dihedral group, a generalized quaternion group, a
P
-group or a cyclic group of order
p
q
n
, where
p
and
q
are distinct primes and
n
is a positive integer.
Let
Γ
be a graph with vertex set
V
. If a subset
C
of
V
is independent in
Γ
and every vertex in
V
\
C
is adjacent to exactly one vertex in
C
, then
C
is called a perfect code of
Γ
. Let
G
be a finite ...group and let
S
be a square-free normal subset of
G
. The Cayley sum graph of
G
with respect to
S
is a simple graph with vertex set
G
and two vertices
x
and
y
are adjacent if
x
y
∈
S
. A subset
C
of
G
is called a perfect code of
G
if there exists a Cayley sum graph of
G
which admits
C
as a perfect code. In particular, if a subgroup of
G
is a perfect code of
G
, then the subgroup is called a subgroup perfect code of
G
. In this paper, we give a necessary and sufficient condition for a non-trivial subgroup of an abelian group with non-trivial Sylow 2-subgroup to be a subgroup perfect code of the group. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian 2-groups. As an application, we classify the abelian groups whose every non-trivial subgroup is a subgroup perfect code. Moreover, we determine all subgroup perfect codes of a cyclic group, a dihedral group and a generalized quaternion group.
A weakly distance-regular digraph is quasi-thin if the maximum value of its intersection numbers is 2. In this paper, we focus on commutative quasi-thin weakly distance-regular digraphs, and classify ...such digraphs with valency more than 3. As a result, this family of digraphs is completely determined.
Integral Cayley Sum Graphs and Groups Ma, Xuanlong; Wang, Kaishun
Discussiones Mathematicae. Graph Theory,
11/2016, Letnik:
36, Številka:
4
Journal Article
Recenzirano
Odprti dostop
For any positive integer k, let A
denote the set of finite abelian groups G such that for any subgroup H of G all Cayley sum graphs CayS(H, S) are integral if |S| = k. A finite abelian group G is ...called Cayley sum integral if for any subgroup H of G all Cayley sum graphs on H are integral. In this paper, the classes A
and A
are classified. As an application, we determine all finite Cayley sum integral groups.
In Levstein and Maldonado (2007), the Terwilliger algebra of the Johnson scheme J(n,d) was determined when n≥3d. In this paper, we determine the Terwilliger algebra T for the remaining case 2d≤n<3d.
For a finite noncyclic group
, let Cyc(
) be the set of elements
of
such that 〈
,
〉 is cyclic for each
of
. The noncyclic graph of
is a graph with the vertex set
∖ Cyc(
), having an edge between two ...distinct vertices
and
if 〈
,
〉 is not cyclic. In this paper, we classify all finite noncyclic groups whose noncyclic graphs are
-free, where
is a star and 3 ≤
≤ 6.
Erigeron multiradiatus (Lindl.) Benth. has been used in Tibet folk medicine to treat various inflammatory diseases. The aim of this study was to investigate antimyocardial ischemia and reperfusion ...(I/R) injury effect of caffeoylquinic acids derivatives of E. multiradiatus (AE) in vivo and to explain underling mechanism. AE was prepared using the whole plant of E. multiradiatus and contents of 6 caffeoylquinic acids determined through HPLC analysis. Myocardial I/R was induced by left anterior descending coronary artery occlusion for 30 minutes followed by 24 hours of reperfusion in rats. AE administration (10, 20, and 40 mg/kg) inhibited I/R-induced injury as indicated by decreasing myocardial infarct size, reducing of CK and LDH activities, and preventing ST-segment depression in dose-dependent manner. AE decreased cardiac tissue levels of proinflammatory factors TNF-α and IL-6 and attenuated leukocytes infiltration. AE was further demonstrated to significantly inhibit I-κB degradation, nuclear translocation of p-65 and phosphorylation of JNK. Our results suggested that cardioprotective effect of AE could be due to suppressing myocardial inflammatory response and blocking NF-κB and JNK activation pathway. Thus, caffeoylquinic acids might be the active compounds in E. multiradiatus on myocardial ischemia and be a potential natural drug for treating myocardial I/R injury.
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