Gen. Math. Notes, Vol. 19, No. 1, November, 2013, pp. 28-34
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ISSN 2219-7184; Copyright ICSRS
Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
Tensor Product of Colour Vertex Transitive
Cayley Graphs of Finite Semigroups
A. Assari1 and N. Hosseinzadeh2
1
Department of Basic Science
Jundi-Shapur University of Technology
Dezful, Iran
E-mail:amirassari@jsu.ac.ir
2
Department of Mathematics
Dezful Branch, Islamic Azad University
Dezful, Iran
E-mail:narges.hosseinzadeh@gmail.com
(Received: 11-8-13 / Accepted: 15-9-13)
Abstract
It is important to see what kinds of properties of graphs can be transfer to
the product of them. Here we will prove that the colour automorphism vertex
transitivity can be transfer, and bring a special proof for vertex transitivity
property of Cayley graphs of semigroups as well.
Keywords: Cayley graph, semigroup, colour automorphism vertex transitive, tensor product, automorphism group
1
Introduction
Let G be a semigroup and S be a nonempty subset of G. The Cayley graph
of G relative to S is denoted by Γ = Cay(G, S) is a graph with vertex set G
and (x, y) is an edge of Γ if and only if for some s ∈ S we have y = sx.
The aim of this paper is to prove that the tensor product of two Cayley
graphs of semigroups preserves the colour automorphism vertex transitivity
condition. Let us first define a few properties of graphs as well as Cayley
graphs.
Tensor Product of Colour Vertex Transitive...
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Let Γ1 = (V1 , E1 ) and Γ2 = (V2 , E2 ) be two graphs. Γ = (V, E), the tensor
product of them is a graph with vertex set V = V1 ×V2 , and ((u1 , u2 ), (v1 , v2 ))is
an edge in Γ if and only if (u1 , v1 ) ∈ E1 and (u2 , v2 ) ∈ E2 .
An endomorphism of a graph Γ = (V, E) is a mapping φ : V −→ V which
preserves the edges of Γ. An automorphism of the graph Γ is a on-to-one and
onto endomorphism of Γ. We denote the set of all endomorphisms (which is
a monoid with the composition operator) and the set of all automorphisms (
which is a group) of Γ by End(Γ) and Aut(Γ) respectively .
A graph Γ = (V, E) is said to be vertex transitive if for any two vertices
x, y ∈ V , there exists an automorphism of Γ which sends x to y. More generally,
a subset X of End(Γ) is said to be vertex transitive on Γ and Γ is said to be Xvertex transitive, if for any two vertices x, y ∈ V , there exist an endomorphism
φ ∈ X which sends x to y .
Let G be a semigroup and S be a subset of G. An element of φ ∈
End(Cay(G, S)) is called colour preserving endomorphism if sx = y implies
s(xφ ) = y φ for every x, y ∈ G and s ∈ S . Denoted by ColEndS (G) and
ColAutS (G) the sets of all colour preserving endomorphisms and colour preserving automorphisms of Cay(G, S) respectively. Evidently,
ColAutS (G) ⊂ Aut((Cay(G, S))
ColEndS (G) ⊂ End(Cay(G, S))
and ColoAutS (G) and ColEndS (G) are submonoids of End(Cay(G, S)). [3]
A Cayley graph Γ = Cay(G, S) is called colour automorphism vertex transitive if it is ColAutS (G)-vertex transitive.
We use the notation of [2] for concepts of semigroups. A band is a semigroup
entirely consisting of idempotents. A right (left) zero semigroup is a semigroup
in which we have xy = y (xy = x) for all elements of the semigroup. A simple
semigroup S is a semigroup with no proper ideal (There exists no A ⊂ S with
A 6= S such that SA ⊂ A and AS ⊂ A). By a completely simple semigroup
we mean a simple semigroup containing a primitive idempotent.
The Cayley graphs of semigroups is a main consideration in the literature,
for example Wang et all in [7] gave necessary and sufficient conditions for
various vertex-transitivity of Cayley graphs of the class of completely 0-simple
semigroups and its several subclasses. Many large graphs can be constructed
by expanding of small graphs, thus it is important to know which properties
of small graphs can be transfer to the expanded one, for example Wang in
[4] proved that the lexicographic of vertex transitive graphs is also vertex
transitive. Michel et all in [5] calculated the distance between two vertices in
Cartesian product of two graphs as well as the diameter of the produced graph
with respect to the primary graphs and used it to find a condition under that
the Cartesian product of two graphs be hyperbolic. It is well known that the
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A. Assari et al.
tensor product of two graphs preserves the transitivity condition. Specapan in
[6] Found the fewest number of vertices for Cartesian product of two graphs
whose removal from the graph results in a disconnected or trivial graph. This
motivated us to consider the Tensor product of colour automorphism Cayley
graphs of semigroups and prove that this kind of product also preserve the
colour automorphism vertex transitivity condition in Cayley graphs of finite
semigroups and bring another proof for the vertex transitivity of such graphs
as well.
2
Preliminary Notes
The tensor product of Cayley graphs of group is also a Cayley graph [1]. We
will show this is also true for the Cayley graphs of semigroups.
Lemma 2.1. Let Γ1 = Cay(H, S) and Γ2 = Cay(K, T ) be two Cayley
graphs of semigroups, then the tensor product of them is also a Cayley graph
of semigroups.
Proof. Set Γ = (V, E) the tensor product of Γ1 and Γ2 . V = H × K is also a
semigroup and ((v1 , v2 ), (u1 , u2 )) ∈ E if and only if (v1 , u1 ) is an edge of Γ1 and
(v2 , u2 ) is an edge of Γ2 , i.e there exist s ∈ S and t ∈ T such that u1 = sv1 and
u2 = tv2 and this will happen if and only if (u1 , u2 ) = (s, t)(v1 , v2 ) for some
(s, t) ∈ S × T . Thus Γ is a Cayley graph of the semigroup H × T relative to
the subset S × T of that.
In [3], the authors provide some sufficient conditions under that a Cayley
graph of a semigroup may be vertex transitive or colour automorphism vertex
transitive which we bring them here.
Theorem 2.2. Let G be a finite semigroup and S ba a subset of G. Then
the Cayley graph Cay(G, S) is colour automorphism vertex transitive if and
only if the following conditions hold:
(i) sG = G, for all s ∈ S;
(ii) < S > is isomorphic to a direct product of a right zero band and a group;
(iiii) | < S > g| is independent of the choice of g ∈ G.
Theorem 2.3. Let G be a finite semigroup and S be a subset of G. Then
the Cayley graph Cay(G, S) is vertex transitive if and only if the following
conditions hold:
(i) sG = G, for all s ∈ S;
Tensor Product of Colour Vertex Transitive...
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(ii) < S > is a completely simple semigroup;
(iii) the Cayley graph Cay(< S >, S) is vertex transitive;
(iv) | < S > g| is independent of the choice of g ∈ G.
3
Main Results
Let H and K be two finite semigroups and S and T be two subsets of them respectively. Set Γ1 = Cay(H, S) and Γ2 = Cay(K, T ) and Γ the tensor product
of them. we will show that the colour automorphism vertex transitivity of them
can transfer to the tensor product of them as well as the vertex transitivity.
Theorem 3.1. Let Γ1 and Γ2 be two colour automorphism vertex transitive
Cayley graphs. Then Γ = (V, E) the tensor product of them is also colour
automorphism vertex transitive.
Proof. By the definition of tensor product and lemma 2.1 we have Γ = Cay(G, W )
where G = H × K and W = S × T . By theorem 2.2 it is enough to show
the three mentioned conditions holds for the semigroup G and its subset W
where we know that these conditions hold for the semigroups H and K and
their subsets S and T .
(i) Let w = (s, t) ∈ W = S × T . By assumptions we have sH = H and
tK = K, implying
wG = (s, t)(H × K) = sH × tK = H × K = G
(ii) By assumptions and theorem 2.2 part (ii), we can write < S > and
< T > in the form of B × M and C × P respectively, where B and C
are two right zero bands and M and P are two groups.
Every element of < S > can be written in the form of Πi∈I si for some
index set I and si ∈ S ⊂< S >= B × M and hence si = (bi , mi ) for some
bi ∈ B and mi ∈ M . B is a right zero band and thus Πi∈I bi = bj for
some j ∈ I, implying Πi∈I si = (bj , Πi∈I mi ) for some bj ∈ B and mi ∈ M .
Therefore we have
B = {bi |∃si ∈ S, ∃mi ∈ M ; si = (bi , mi )}
and we observe that the cardinality of the set B is not bigger than the
cardinality of the set S.
Similarly we can say that
C = {ci |∃ti ∈ T, ∃pi ∈ P ; ti = (ci , pi )}
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A. Assari et al.
and every element of < T > can be written in the form of Πj∈J tj =
(ck , Πj∈J pj ) where J is an index set and ck ∈ C for some k ∈ J and
pj ∈ P for all j ∈ J.
Now suppose w ∈< S × T >, thus w = Πi∈I (si , ti ) for some si ∈ S and
ti ∈ T . Since S and T are subsets of < S > and < T > respectively, we
can write w by the product of four tuple (bi , mi , ci , pi ) which is isomorphic
to the product of four tuple (bi , ci , mi , pi ) for i ∈ I, bi ∈ B, ci ∈ C, mi ∈ M
and pi ∈ P . But B and C are right zero band, implying we can write w
in the form (bk , ck , Πı∈I mi , Πi∈I pi ) for some k ∈ I. Implying < S × T >
is a subset of (B × C) × (M × P ).
Now we want to prove that the converse is also true, i.e. (B×C)×(M ×P )
is also a subset of < S × T >. Suppose (b, c, m, p) be an element of
(B × C) × (M × P ). m ∈ M and p ∈ P implying there exists some
bm ∈ B, an index set I, si ∈ S∀i ∈ I, cp ∈ C and an index set J and
tj ∈ P ∀j ∈ J such that Πi∈I si = (bm , m) and Πj∈J tj = (cp , p). Since
(b, 1M ) ∈< S >= B × M and (c, 1P ) ∈< T >= C × P and being right
zero bandness of B and C, one can deduce Πj∈J + (si , ti ) = (b, m, c, p) ∼
=
(b, c, m, p) for some Index sex J + which include I and J. Hence we have
< S × T >∼
= (B × C) × (M × P )
where B × C is a right zero band and M × P is a group.
(iii) By assumption | < S > h| is independent of h ∈ H as well as | <
T > k| relative to k ∈ K. Mapping φ :< S > h →< S > h0 for
h, h0 ∈ H which sends Πi∈I si h to Πi∈I si h0 is onto with the condition
| < S > h| = | < S > h0 | which is finite, implies that φ is a bijection.
Similarly happens for the mapping ψ :< T > k →< T > k 0 for k, k 0 ∈ K
defined by ψ(Πj∈J tj k) = Πj∈J tj k 0 . Implying the mapping (φ, ψ) :<
S × T > (h, k) →< S × T > (h0 , k 0 ) which sends Πi∈I (si , ti )(h, k) to
Πi∈I (si , ti )(h0 , k 0 ) is a bijection, i.e. | < S × T > (h, k)| is independent of
(h, k) ∈ H × K.
It is a well known that the product of two vertex transitive graphs is also
vertex transitive. But we bring an innovative method to proof it for vertex
transitive Cayley graphs of semigroups which is a special case of it.
Theorem 3.2. Let Γ1 and Γ2 be two vertex transitive Cayley graphs of
finite semigroups. Then Γ = (V, E) the tensor product of them is also vertex
transitive Cayley graph.
Tensor Product of Colour Vertex Transitive...
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Proof. We bring the notation of theorem 3.1 and by the theorem 2.3 it is
sufficient to prove that G satisfies the four conditions in the theorem 2.3.
Conditions (i) and (iv) is proved in the proof of theorem 3.1. Thus is it only
to verify the conditions (ii) and (iii).
(ii) By lemma 2.1, we have to show < S × T > is a completely simple
semigroup.
For simplicity of < S × T >, suppose not and let A be a proper ideal
of < S × T >. Let B = π1 (A) and C = π2 (A) which are ideal of
< S > and < T > respectively, and since A is a proper ideal, at least
one of B or C should be a proper ideal of < S > or < T > respectively,
which is contradiction to the hypothesis, implying < S × T > is a simple
semigroup.
Now by theorem 2.3, < S > has a primitive element such as e1 and
< T > also has a primitive element such as e2 . one can verify that
e = (e1 , e2 ) is a primitive element of < S × T >.
Thus < S × T > is a completely simple semigroup.
(iii) By theorem 2.3(iii), Γ1 = Cay(< S >, S) and Γ2 = Cay(< T >, T ) are
vertex transitive, thus for two arbitrary vertices of Γ = Cay(< S × T >
, S × T ) such as a = (ΠI si , ΠI ti ) and b = (ΠJ s0j , ΠJ t0j ), there exists
σ ∈ Aut(Γ1 ) and δ ∈ Aut(Γ2 ) such that σ sends ΠI si to ΠJ s0j and δ
sends ΠI ti to ΠJ t0j . Since (σ, δ) ∈ Aut(Γ) sends a to b yields Γ is a vertex
transitive.
References
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