ON CONNECTEDNESS AND COMPLETENESS OF CAYLEY DIGRAPHS OF TRANSFORMATION SEMIGROUPS WITH FIXED SETS Nuttawoot Nupo and Chollawat Pookpienlert

Let Fix(X,Y ) be a semigroup of full transformations on a set X in which elements in a nonempty subset Y of X are fixed. In this paper, we construct the Cayley digraphs of Fix(X,Y ) and study some structural properties of such digraphs such as the connectedness and the completeness. Further, some prominent results of Cayley digraphs of Fix(X,Y ) relative to minimal idempotents are verified. In addition, the characterization of an equivalence digraph of the Cayley digraph of Fix(X,Y ) is also investigated. Mathematics Subject Classification (2020): 05C20, 05C25, 20M20


Introduction
In algebraic graph theory, the structures of algebraic methods are studied and then applied to problems about graphs. An interesting topic is to study properties of graphs in connection to algebraic systems. A well-known connection between graphs and algebraic systems is the construction of Cayley graphs of groups. The Cayley graph was first introduced for finite groups by Arthur Cayley in 1878. This concept was considered to explain the structures of abstract groups which are described by the set of group generators. Furthermore, the construction of Cayley graphs is also applied to semigroups. As the fact that Cayley graphs of semigroups can reflect the structural properties of semigroups, such semigroups can be visualized by constructing their Cayley graphs. For introducing the definition of the Cayley graph, let S be a semigroup and A a subset of S. The Cayley graph Cay(S, A) of a semigroup S with respect to A is defined to be a digraph with vertex set S and arc set consisting of ordered pairs (x, xa) for some a ∈ A and x is an arbitrary element in S. The set A is called a connection set of Cay(S, A). It is easily visible that if A is an empty set, then Cay(S, A) is considered to be an empty graph.
Thus throughout this paper, the connection set A will be nonempty. The Cayley Hao and Clarke [11] considered Cayley graphs of completely simple semigroups.
In addition, they studied some structural properties such as the completeness and strongly connected bipartite Cayley graphs. Indeed, it turns out that Cayley graphs of semigroups are significant not only in semigroup theory, but also in constructions of various interesting types of graphs with nice combinatorial properties. Several prominent properties of those graph constructions have been presented in numerous journals. In 2015, Suksumran and Panma [16] proposed some concepts on connected Cayley graphs of semigroups. Later in 2016, Afkhami et al. [1] constructed a new class of Cayley graphs and studied their structural properties similar to the research presented by Sinha and Sharma [15] in the same year. Furthermore, in 2018, Panda and Krishna [12] investigated the connectedness of power graphs of finite groups.
For the part of semigroup theory, one of well-known semigroups that plays a crucial role in the study of semigroups is a transformation semigroup. In group theory, the well-known theorem called Cayley's theorem asserts that any group G is isomorphic to a subgroup of the symmetric group of the set G. Also in semigroup theory, an analogue of Cayley's theorem shows that any semigroup can be realized as a transformation semigroup of certain appropriate set. This would be a general result why the transformation semigroup is interesting to study. Some prominent results of Cayley graphs of transformation semigroups have been obtained. For instance, in 2017, Tisklang and Panma [17] investigated the connectedness of Cayley graphs of finite transformation semigroups with restricted range. Moreover, Riyas and Geetha [14], in 2018, studied the Cayley graphs of full transformation semigroups relative to the sets of idempotents. They also provided the existence of Hamiltonian cycles in such Cayley graphs. Our purpose is to study some structural properties of Cayley graphs of certain transformation semigroups. Let X be a set and Y a nonempty subset of X. Further, let T (X) denote the semigroup of 112 NUTTAWOOT NUPO AND CHOLLAWAT POOKPIENLERT transformations from X into itself under the composition of maps which is generally called the full transformation semigroup. Define the transformation semigroup Fix(X, Y ) with a fixed set Y as follows: Then Fix(X, Y ) is a subsemigroup of T (X). Virtually, whenever Y has only one element, say a, the semigroup we consider is the set of all self-maps on X having a as their only common fixed point which is the one of interesting topics studied in fixed point theory.
The semigroup Fix(X, Y ) was first introduced by Honyam and Sanwong [7] in 2013. They characterized the regularity, Green's relations and ideals of Fix(X, Y ).
Later in 2016, Chaiya, Honyam and Sanwong [2] presented the characterization of the natural partial order on Fix(X, Y ).
Here, we shall investigate certain classes of Cayley graphs of Fix(X, Y ) such as connected digraphs, complete digraphs and equivalence digraphs. Some useful notations and relevant terminologies related to this paper will be provided in the next section.

Preliminaries and notations
In this section, some basic preliminaries and relevant notations used in what follows on digraphs, semigroups and Cayley graphs of semigroups are described.
All sets mentioned in this paper are assumed to be finite.
then D is said to be a loopless digraph.
Moreover, the digraph D is said to be semi-complete if for every u, v ∈ V (D), Let D be a digraph. Consider a sequence P of distinct vertices in V (D) as follows: ON CONNECTEDNESS AND COMPLETENESS OF CAYLEY DIGRAPHS 113 P := v 1 , v 2 , . . . , v k for some k ∈ N.
If P satisfies the condition that either Moreover, if P satisfies that (v i , v i+1 ) ∈ E(D) for all i = 1, 2, . . . , k − 1, then P is said to be a dipath from v 1 to v k in D. For convenience, throughout this paper, the Let X be a set and Y a nonempty subset of X. For convenience, we let Y = {a i : i ∈ I} throughout this paper, unless otherwise stated. Hence for each α ∈ Fix(X, Y ), we observe that a i α = a i for all i ∈ I. According to the convention presented in [7], we now give a presentation for the elements of Fix(X, Y ). For each and take as understood that the subscripts i and j belong to the index sets I and J, respectively. Moreover, Xα, the image of α, is the disjoint union of Actually, the index set J can be empty in general. Furthermore, the notation π α stands for the set {xα −1 : x ∈ Xα}.
Let A and B be families of sets. If for each A ∈ A , there exists B ∈ B in which A ⊆ B, we say that A refines B. It is not hard to prove the property stated in the following lemma.
In this paper, we investigate the properties of the connectedness and the com- An idempotent e is said to be minimal if e has the property f ∈ E(S) and f ≤ e implies f = e.

Generally, it is well known that α ∈ T (X) is an idempotent if and only if xα = x
for all x ∈ Xα. Consequently, we obtain that is an idempotent if and only if b j ∈ B j for all j ∈ J.
Further, let E m be a set of all minimal idempotents of Fix(X, Y ). We can conclude from [7] that E m is not empty and it precisely contains all idempotents whose images coincide with Y , that is, Moreover, we consider the property that how any minimal idempotent plays a role in the construction of the Cayley graph of Fix(X, Y ) with respect to a connection set A.
First of all, we note that every minimal idempotent is a left zero element of Then µ is written as follows:
As the above consequence, we directly obtain that Cay(Fix(X, Y ), A) always Let α be any element in Fix(X, Y ). We define As µ ∈ E m is a left zero, all the arcs from µ will form loops and hence we have the following lemma.
We now describe the Cayley graph of Fix(X, Y ) whose connection set is a sin- is a disjoint union of n subdigraphs which each of them contains exactly one minimal idempotent as its vertex.
and α a vertex of C. Then we can write Since µ ∈ E m , µ is also expressed as C kj for some j ∈ J} and I = I\K. Then Since (α, αµ) ∈ E(Cay(Fix(X, Y ), A)), we obtain that αµ lies in C. Hence C contains a minimal idempotent αµ as its vertex.
Next, we will show that C contains exactly one minimal idempotent as its vertex.
Suppose that µ 1 and µ 2 are different minimal idempotents contained in V (C). Thus By the definition of a semidipath, all vertices occurred in the above expression of the [µ 1 , µ 2 ]-semidipath must be distinct. Since  Proof. Assume that C is the component of Cay(Fix(X, Y ), A) containing η. For each α ∈ V (C), we have that αµ is a minimal idempotent so that αµ = η. Hence any arc in C is of the form (α, η).

Connectedness of Cay(Fix(X, Y ), A)
In this section, we provide results on connectedness of Cay(Fix(X, Y ), A) consisting of the strongly connectedness, weakly connectedness, locally connectedness, and unilaterally connectedness.  Conversely, suppose that Y is a proper subset of X. It is not hard to verify that Then µ ∈ E m . By Lemma 3.1, we have N + (µ) = ∅. That means there is no arc joining from µ to any element in Fix(X, Y )\{µ}. Consequently, there is no dipath from µ to another vertex. Therefore, Cay(Fix(X, Y ), A) is not strongly connected.
This completes the proof.
Before we present the weakly connectedness of Cay(Fix(X, Y ), A), we need to prescribe the special notation as follows. Let A be a nonempty subset of Fix(X, Y ), the notation A 1 stands for the set A adjoined the identity id X of Fix(X, Y ), that is, is weakly connected if and only if one of the following conditions holds: Proof. Assume that Cay(Fix(X, Y ), A) is weakly connected. We now suppose that Y is a proper subset of X. Then |Fix(X, Y )| ≥ 2. Next, let α, β ∈ Fix(X, Y ) be such that α = β.
Conversely, if X = Y , then Cay(Fix(X, Y ), A) is weakly connected since it is strongly connected as shown in Theorem 4.1. Next, we assume that the second condition holds. Let α, β ∈ Fix(X, Y ) be such that α = β. Then there exist for all i = 0, 1, 2, . . . , k where α 0 = α and α k+1 = β. Thus for each i, there exists We now have two possibilities to investigate.
Case 1: either δ or σ is an identity. It is easily seen that there exists a directed edge joining between α i and α i+1 .
Since aµ = a = aβ for all a ∈ Y and we have bµ = bβ, it follows that µ = β ∈ A which leads to E m ⊆ A.
(iii) ⇒ (i) Let E m be a subset of A and α, β ∈ Fix(X, Y ). It is obvious that Therefore, we can find an [α, β]-semidipath in Cay(Fix(X, Y ), A) which implies that For characterizing the locally connectedness of Cay(Fix(X, Y ), A), we need the following lemma. Proof. Assume that there exists a dipath P joining from α to β in Cay(Fix(X, Y ), A).
Further, the notation H id X means an equivalence H-class containing id X and elements in Fix(X, Y ) which H-relate to id X where H is one of Green's relations (see [8]). Moreover, it is well known that H id X is a group of all bijections, exactly. 4.4, we can conclude that π α refines π id X and so α is injective. Since X is finite, we obtain that α is a bijection, that is, α ∈ H id X . Hence A ⊆ H id X , as required.
Conversely, assume that A ⊆ H id X . Let α, β ∈ Fix(X, Y ) be such that Cay (Fix(X, Y ), A) contains an [α, β]-dipath. Then there exist bijections λ 1 , λ 2 , . . . , λ k ∈ A ⊆ H id X for some k ∈ N in which β = αλ 1 λ 2 · · · λ k . Without loss of generality, we may assume that such λ i is not an identity of Fix(X, Y ) for all i = 1, 2, . . . , k. Since H id X is a finite group, we can define the order of λ i as r i where r i > 1 for all i = 1, 2, . . . , k, that is, λ ri i is an identity of H id X which coincides with the identity of Fix(X, Y ). Hence This ensures that there exists a dipath from β to α. So Cay(Fix(X, Y ), A) is locally connected. is unilaterally connected if and only if one of the following conditions holds: (ii) |X| = 2, |Y | = 1 and A contains a minimal idempotent.
We consequently obtain that there is no dipath joining between µ 1 and µ 2 which contradicts to the unilaterally connectedness of Cay(Fix(X, Y ), A). Thus |Y | = 1. where γ is the unique minimal idempotent of Fix(X, Y ).

NUTTAWOOT NUPO AND CHOLLAWAT POOKPIENLERT
Suppose that A does not contain a minimal idempotent. So A = {id X } and Cay(Fix(X, Y ), A) is shown in Figure 3. We observe that Cay(Fix(X, Y ), A) is not unilaterally connected which is impossible. Hence the connection set A of Cay(Fix(X, Y ), A) must contain a minimal idempotent. Therefore, the necessity is completely proved.
γ id X  Clearly, both Cayley graphs are unilaterally connected.

Completeness of Cay(Fix(X, Y ), A)
In this section, we study characterizations of the completeness of Cay(Fix(X, Y ), A).
Besides the completeness of Cay(Fix(X, Y ), A), the concept of an equivalence digraph is also considered for Cay(Fix(X, Y ), A). We now present the following results. is complete if and only if X = Y .
Proof. It is obvious that Cay(Fix(X, Y ), A) is complete whenever X = Y since it contains exactly one vertex and a loop attached to the vertex.
To prove the converse, assume that Cay(Fix(X, Y ), A) is complete. Suppose to the contrary that there exists b ∈ X\Y . Choose a i0 ∈ Y and let I = I\{i 0 }. Define Clearly, α ∈ E m and β / ∈ E m . By the completeness of Cay(Fix(X, Y ), A), we have (α, β), (β, α) ∈ E(Cay(Fix(X, Y ), A)). The edge (α, β) implies that there exists λ ∈ A in which β = αλ = α since α ∈ E m . This gives a contradiction. So we can conclude that X = Y .
is semi-complete if and only if one of the following conditions holds: (ii) |X| = 2, |Y | = 1 and A = Fix(X, Y ).
Proof. Let Cay(Fix(X, Y ), A) be semi-complete. Suppose that Y is a proper subset of X. Similarly, we can apply the proof of the necessity of Theorem 4.6 to conclude that |X| = 2 and |Y | = 1. Consequently, Fix(X, Y ) = {id X , γ} and γ ∈ E m . If id X / ∈ A, then (id X , id X ) / ∈ E(Cay(Fix(X, Y ), A)) or if γ / ∈ A, then (γ, id X ), (id X , γ) / ∈ E(Cay(Fix(X, Y ), A)), which contradicts the semicompleteness of Cay(Fix(X, Y ), A). Hence both of id X and γ must belong to A which implies that A = Fix(X, Y ).
Conversely, assume that the second condition holds. Thus Cay(Fix(X, Y ), A) can be drawn as Figure 5 which is shown in the proof of the sufficiency of Theorem 4.6.
So Cay(Fix(X, Y ), A) is semi-complete. Moreover, the assertion is clear whenever Note that Fix(X, Y ) always contains minimal idempotents. Since they are left zeros of Fix(X, Y ), the digraph Cay(Fix(X, Y ), A) always contains loops attached to the minimal idempotents. Consequently, Cay(Fix(X, Y ), A) is not a loopless digraph. Hence, we have the following theorem. is never a directed complete digraph.

NUTTAWOOT NUPO AND CHOLLAWAT POOKPIENLERT
We now present the following results on an equivalence digraph for Cay(Fix(X, Y ), A).
Lemma 5.4. Let A be a nonempty subset of Fix(X, Y ). Then E(Cay (Fix(X, Y ) is reflexive if and only if id X ∈ A.
Proof. Let E(Cay (Fix(X, Y ), A)) be a reflexive relation. Since Fix(X, Y ) contains the identity id X , we obtain that (id X , id X ) ∈ E(Cay(Fix(X, Y ), A)). Thus there exists α ∈ A such that id X = id X α = α ∈ A.
For the converse, it is easy to verify that (α, α) ∈ E(Cay(Fix(X, Y ), A)) for all α ∈ Fix(X, Y ) whenever A contains an identity of Fix(X, Y ). We obtain that E(Cay(Fix(X, Y ), A)) is reflexive, as required.
Recall that H id X is a group with identity id X . For a nonempty subset B of H id X , the notation B −1 stands for the set {δ −1 : δ ∈ B} which is useful for proving the symmetry of E(Cay(Fix(X, Y ), A)) stated in the following lemma. Proof. Assume that E(Cay(Fix(X, Y ), A)) is symmetric. Clearly, Cay(Fix(X, Y ), A) is locally connected. By Theorem 4.5, we have A ⊆ H id X . Further, let α ∈ A.
Consequently, we have A = A −1 .
From Lemmas 5.4, 5.5 and 5.6, we directly have the following theorem. is an equivalence digraph if and only if A is a subgroup of H id X .