scholarly journals On extendability of Cayley graphs

Filomat ◽  
2009 ◽  
Vol 23 (3) ◽  
pp. 93-101 ◽  
Author(s):  
Stefko Miklavic ◽  
Primoz Sparl
Keyword(s):  
Abelian Group ◽  
Cayley Graph ◽  
Connected Graph ◽  
Perfect Matching ◽  
Cayley Graphs ◽  
Dihedral Groups ◽  
Odd Order ◽  
Even Order

A connected graph ? of even order is n-extendable, if it contains a matching of size n and if every such matching is contained in a perfect matching of ?. Furthermore, a connected graph ? of odd order is n1/2-extendable, if for every vertex v of ? the graph ? - v is n-extendable. It is proved that every connected Cayley graph of an abelian group of odd order which is not a cycle is 1 1/2-extendable. This result is then used to classify 2-extendable connected Cayley graphs of generalized dihedral groups.

scholarly journals Which Haar graphs are Cayley graphs?

10.37236/5240 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
István Estélyi ◽  
Tomaž Pisanski
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Cayley Graph ◽  
Regular Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Equivalent Condition ◽  
Dihedral Groups ◽  
A Finite Group

For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$ is well-known to be a Cayley graph; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph $H(G,S)$ is a Cayley graph. An equivalent condition for $H(G,S)$ to be a Cayley graph of a group containing $G$ is derived in terms of $G, S$ and $\mathrm{Aut } G$. It is also shown that the dihedral groups, which are solutions to the above problem, are $\mathbb{Z}_2^2,D_3,D_4$ and $D_{5}$. 

scholarly journals Integral Cayley Graphs over Abelian Groups

10.37236/353 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Additive Group ◽  
Cyclic Groups ◽  
Vertex Set ◽  
A Finite Group ◽  
Gamma S

Let $\Gamma$ be a finite, additive group, $S \subseteq \Gamma, 0\notin S, -S=\{-s: s\in S\}=S$. The undirected Cayley graph Cay$(\Gamma,S)$ has vertex set $\Gamma$ and edge set $\{\{a,b\}: a,b\in \Gamma$, $a-b \in S\}$. A graph is called integral, if all of its eigenvalues are integers. For an abelian group $\Gamma$ we show that Cay$(\Gamma,S)$ is integral, if $S$ belongs to the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. The converse is proven for cyclic groups. A finite group $\Gamma$ is called Cayley integral, if every undirected Cayley graph over $\Gamma$ is integral. We determine all abelian Cayley integral groups.

MATCHING PROPERTIES IN DOUBLE DOMINATION EDGE CRITICAL GRAPHS

2010 ◽  
Vol 02 (02) ◽  
pp. 151-160 ◽  
Author(s):  
HAICHAO WANG ◽  
LIYING KANG
Keyword(s):  
Perfect Matching ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Critical Graphs ◽  
Critical Graph ◽  
Odd Order ◽  
Even Order ◽  
Double Domination

A vertex subset S of a graph G = (V, E) is a double dominating set for G if |N[v]∩S| ≥ 2 for each vertex v ∈ V, where N[v] = {u |uv ∈ E}∪{v}. The double domination number of G, denoted by γ×2(G), is the cardinality of a smallest double dominating set of G. A graph G is said to be double domination edge critical if γ×2(G + e) < γ×2(G) for any edge e ∉ E. A double domination edge critical graph G with γ×2(G) = k is called k - γ×2(G)-critical. In this paper, we first show that G has a perfect matching if G is a connected 3 - γ×2(G)-critical graph of even order. Secondly, we show that G is factor-critical if G is a connected 3 - γ×2(G)-critical graph with odd order and minimum degree at least 2. Finally, we show that G is factor-critical if G is a connected K1,4-free 4 - γ×2(G)-critical graph of odd order with minimum degree at least 2.

scholarly journals 4-Factor-Criticality of Vertex-Transitive Graphs

10.37236/4807 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Wuyang Sun ◽  
Heping Zhang
Keyword(s):  
Perfect Matching ◽  
Transitive Graph ◽  
Critical Graphs ◽  
Odd Order ◽  
Even Order ◽  
Vertex Transitive ◽  
P Factor ◽  
Vertex Transitive Graph ◽  
Transitive Graphs ◽  
Connected Vertex

A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-factor-critical graphs and 2-factor-critical graphs are well-known factor-critical graphs and bicritical graphs, respectively. It is known that if a connected vertex-transitive graph has odd order, then it is factor-critical, otherwise it is elementary bipartite or bicritical. In this paper, we show that a connected vertex-transitive non-bipartite graph of even order at least 6 is 4-factor-critical if and only if its degree is at least 5. This result implies that each connected non-bipartite Cayley graph of even order and degree at least 5 is 2-extendable.

scholarly journals Integral Cayley Graphs Defined by Greatest Common Divisors

10.37236/581 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Direct Product ◽  
Undirected Graph ◽  
Cayley Graphs ◽  
Cyclic Groups ◽  
Greatest Common Divisors ◽  
Gamma S

An undirected graph is called integral, if all of its eigenvalues are integers. Let $\Gamma =Z_{m_1}\otimes \ldots \otimes Z_{m_r}$ be an abelian group represented as the direct product of cyclic groups $Z_{m_i}$ of order $m_i$ such that all greatest common divisors $\gcd(m_i,m_j)\leq 2$ for $i\neq j$. We prove that a Cayley graph $Cay(\Gamma,S)$ over $\Gamma$ is integral, if and only if $S\subseteq \Gamma$ belongs to the the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. It is also shown that every $S\in B(\Gamma)$ can be characterized by greatest common divisors.

scholarly journals EXTENSION OF AUTOMORPHISMS OF SUBGROUPS

2012 ◽  
Vol 54 (2) ◽  
pp. 371-380
Author(s):  
G. G. BASTOS ◽  
E. JESPERS ◽  
S. O. JURIAANS ◽  
A. DE A. E SILVA
Keyword(s):  
Abelian Group ◽  
Finite Groups ◽  
Dihedral Group ◽  
Abelian Groups ◽  
Alternating Group ◽  
Finite Abelian Groups ◽  
Quaternion Group ◽  
Icosahedral Group ◽  
Odd Order ◽  
Even Order

AbstractLet G be a group such that, for any subgroup H of G, every automorphism of H can be extended to an automorphism of G. Such a group G is said to be of injective type. The finite abelian groups of injective type are precisely the quasi-injective groups. We prove that a finite non-abelian group G of injective type has even order. If, furthermore, G is also quasi-injective, then we prove that G = K × B, with B a quasi-injective abelian group of odd order and either K = Q8 (the quaternion group of order 8) or K = Dih(A), a dihedral group on a quasi-injective abelian group A of odd order coprime with the order of B. We give a description of the supersoluble finite groups of injective type whose Sylow 2-subgroup are abelian showing that these groups are, in general, not quasi-injective. In particular, the characterisation of such groups is reduced to that of finite 2-groups that are of injective type. We give several restrictions on the latter. We also show that the alternating group A5 is of injective type but that the binary icosahedral group SL(2, 5) is not.

scholarly journals Distance Powers and Distance Matrices of Integral Cayley Graphs over Abelian Groups

10.37236/2369 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Abelian Group ◽  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Integral Spectrum ◽  
Distance Matrices

It is shown that distance powers of an integral Cayley graph over an abelian group $\Gamma$ are again integral Cayley graphs over $\Gamma$. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.

scholarly journals Structural Properties of Connected Domination Critical Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2568
Author(s):  
Norah Almalki ◽  
Pawaton Kaemawichanurat
Keyword(s):  
Structural Properties ◽  
Perfect Matching ◽  
Minimum Degree ◽  
Domination Number ◽  
Free Graph ◽  
Critical Graphs ◽  
Critical Graph ◽  
Odd Order ◽  
Connected Domination Number ◽  
Even Order

A graph G is said to be k-γc-critical if the connected domination number γc(G) is equal to k and γc(G+uv)<k for any pair of non-adjacent vertices u and v of G. Let ζ be the number of cut vertices of G and let ζ0 be the maximum number of cut vertices that can be contained in one block. For an integer ℓ≥0, a graph G is ℓ-factor critical if G−S has a perfect matching for any subset S of vertices of size ℓ. It was proved by Ananchuen in 2007 for k=3, Kaemawichanurat and Ananchuen in 2010 for k=4 and by Kaemawichanurat and Ananchuen in 2020 for k≥5 that every k-γc-critical graph has at most k−2 cut vertices and the graphs with maximum number of cut vertices were characterized. In 2020, Kaemawichanurat and Ananchuen proved further that, for k≥4, every k-γc-critical graphs satisfies the inequality ζ0(G)≤mink+23,ζ. In this paper, we characterize all k-γc-critical graphs having k−3 cut vertices. Further, we establish realizability that, for given k≥4, 2≤ζ≤k−2 and 2≤ζ0≤mink+23,ζ, there exists a k-γc-critical graph with ζ cut vertices having a block which contains ζ0 cut vertices. Finally, we proved that every k-γc-critical graph of odd order with minimum degree two is 1-factor critical if and only if 1≤k≤2. Further, we proved that every k-γc-critical K1,3-free graph of even order with minimum degree three is 2-factor critical if and only if 1≤k≤2.

scholarly journals Normal edge-transitive and \frac{1}{2}-arc-transitive cayley graphs on non-abelian groups of odd order 3pq, p and q are primes

2018 ◽  
Vol 49 (3) ◽  
pp. 183-194
Author(s):  
Ali Reza Ashrafi ◽  
Bijan Soleimani
Keyword(s):  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Prime Numbers ◽  
Graph Of Groups ◽  
Odd Order ◽  
Edge Transitive

Suppose $p$ and $q$ are odd prime numbers. In this paper, the connected Cayley graph of groups of order $3pq$, for primes $p$ and $q$, are investigated and all connected normal $\frac{1}{2}-$arc-transitive Cayley graphs of group of these orders will be classified.

Isomorphisms of bi-Cayley graphs on Dihedral groups

2020 ◽  
Vol 12 (04) ◽  
pp. 2050051
Author(s):  
Majid Arezoomand ◽  
Afshin Behmaram ◽  
Mohsen Ghasemi ◽  
Parivash Raeighasht
Keyword(s):  
Bipartite Graph ◽  
Cayley Graph ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Dihedral Groups ◽  
Vertex Set

For a group [Formula: see text] and a subset [Formula: see text] of [Formula: see text] the bi-Cayley graph BCay[Formula: see text] of [Formula: see text] with respect to [Formula: see text] is the bipartite graph with vertex set [Formula: see text] and edge set [Formula: see text]. A bi-Cayley graph BCay[Formula: see text] is called a BCI-graph if for any bi-Cayley graph BCay[Formula: see text], [Formula: see text] implies that [Formula: see text] for some [Formula: see text] and [Formula: see text]. A group [Formula: see text] is called a [Formula: see text]-BCI-group if all bi-Cayley graphs of [Formula: see text] with valency at most [Formula: see text] are BCI-graphs. In this paper, we characterize the [Formula: see text]-BCI dihedral groups for [Formula: see text]. Also, we show that the dihedral group [Formula: see text] ([Formula: see text] is prime) is a [Formula: see text]-BCI-group.

Sign in / Sign up

Export Citation Format

Share Document