Let G be a finite group and let k⩾2 be an integer. A (G,k,1)‐difference matrix (DM) is a k×∣G∣ matrix D=(dij) with entries from G, such that for all distinct rows x and y, the multiset of differences ...dxjdyj−1:1⩽j⩽∣G∣ contains each element of G exactly once. Let H be a finite abelian group and let D2H=〈H,b∣b2=1,bhb=h−1,h∈H〉 be the generalized dihedral group of H. It is proved that a (D2H,4,1)‐DM exists if and only if H is of even order and H is not isomorphic to Z4. Also for the nine non‐abelian groups G of order 16, we obtain (G,6,1)‐DMs over five of them, (G,5,1)‐DMs over three of them and a (G,4,1)‐DM over one of them. No (G,5,1)‐DM exists for this last group, where G=Q16.
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An independent subset
of the vertex set
of the graph Γ is an
for Γ if each vertex
∈
has precisely one neighbour in
. In this article, we classify the connected cubic Cayley graphs on generalized ...dihedral groups which admit an efficient dominating set.
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Suppose $G$ is a finite non-abelian group and $\Gamma(G)$ is a simple graph with the non-central conjugacy classes of $G$ as its vertex set. Two different non-central conjugacy classes $A$ and $B$ ...are assumed to be adjacent if and only if there are elements $a,b \in G$ such that $a \in A$, $b \in B$ and $ab = ba$. This graph is called the commuting conjugacy class graph of G. In this paper, the structure of the commuting conjugacy class graph of the generalized dihedral group $D_{(m,n)}$ and the generalized dicyclic group $Dic (A, y, x)$ are completely determined.
Finite groups with very few character values are characterized. The following is the main result of this article: A finite non-abelian group has precisely four character values if and only if it is ...the generalized dihedral group of a non-trivial elementary abelian 3-group. The proof involves the analysis of the centralizers of involutions.
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Let
F
q
be a finite field with
q
elements,
D
2
n
,
r
a generalized dihedral group with
gcd
(
2
n
,
q
)
=
1
, and
F
q
D
2
n
,
r
a generalized dihedral group algebra. Firstly, an explicit expression ...for primitive idempotents of
F
q
D
2
n
,
r
is determined, which extends the results of Brochero Martínez (Finite Fields Appl 35:204–214, 2015). Secondly, all linear complementary dual (LCD) codes and self-orthogonal codes in
F
q
D
2
n
,
r
are precisely described and counted. Some numerical examples are also presented to illustrate our main results.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
A graph is said to be integral (resp. distance integral) if all the eigenvalues of its adjacency matrix (resp. distance matrix) are integers. Let
H
be a finite abelian group, and let
H
=
⟨
H
,
b
|
b
...2
=
1
,
b
h
b
=
h
-
1
,
h
∈
H
⟩
be the generalized dihedral group of
H
. The contribution of this paper is threefold. Firstly, based on the representation theory of finite groups, we obtain a necessary and sufficient condition for a Cayley graph over
H
to be integral, which naturally contains the main results obtained in Lu et al. (J Algebr Comb 47:585–601, 2018). Secondly, a closed-form decomposition formula for the distance matrix of Cayley graphs over any finite groups is derived. As applications, a necessary and sufficient condition for the distance integrality of Cayley graphs over
H
is displayed. Some simple sufficient (or necessary) conditions for the integrality and distance integrality of Cayley graph are exhibited, respectively, from which several infinite families of integral and distance integral Cayley graphs over
H
are constructed. And lastly, some necessary and sufficient conditions for the equivalence of integrity and distance integrity of Cayley graphs over generalized dihedral groups are obtained.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
In this paper, we first prove that the connected cubic edge-transitive bi-Cayley graphs over a generalized dihedral group have girth 6. Using this, a complete classification is given of these graphs.
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