A survey on enhanced power graphs of ﬁnite groups

We survey known results on enhanced power graphs of ﬁnite groups. Open problems, questions and suggestions for future work are also included.


Introduction
The study of graphical representations of algebraic structures, especially groups, has been an energizing and fascinating research area originating from the pioneering paper by Arthur Cayley [26] with many recent results (cf. [11,14,40,54,79]). In particular, graphs associated to groups and other algebraic constructions have valuable applications (cf. [47,53]) and are related to automata theory (cf. [48]). The majority of these types of graphs have natural associated labelings. Therefore, this direction can also be considered as a part of the broader field of research -the investigation of graph labelings. For more information, refer to examples of publications devoted to graph labelings ( [10,46,57,66,71]).
The power graphs of groups are a fairly recent development in the realm of graphs from groups. The power graph of a group was introduced by Kelarev and Quinn [49]. For a group G, the directed power graph of G is denoted by − → P (G) and is defined as a digraph with vertex set G and there is a directed edge from x to y, for x, y ∈ G, x = y, if and only if y is a power of x, i.e., y = x k for some integer k, which is also equivalent to saying that y belongs to the subgroup x generated by x in G. The power graphs of semigroups were investigated in [50,51,52]. All the above papers [49,50,51,52] use the brief term "power graphs" to refer to the directed power graphs.
In 2009, Chakrabarty et al. [27] explored the undirected power graphs of semigroups. The undirected power graph P(G) of a group G coincides with the underlying undirected graph of the directed power graph defined in [49]. It has the vertex set G where two distinct elements are adjacent if one of them is a power of the other. Cameron et al. [22,24] examined the undirected power graphs of finite groups and initiated the use of the brief term "power graph" with the second meaning of an undirected power graph.
Notice that the survey [2] introduced the most general definition of a power graph defined for all power-associative magmas, i.e., sets with a power-associative operation (Definition 1 on p. 126 in [2]). It is well known that magmas play essential roles in symbolic computation (cf. [76,77,78]). Let us refer to two surveys [2,55] for more information pertaining to the research results and open problems on the power graphs of groups.
Enhanced power graphs are another important class of graphs that have been actively investigated recently. The term enhanced power graph of a group was introduced by Aalipour et al. [1] as a graph "lying in between" the power graph and the commuting graph, where the commuting graph of a group G, denoted by C(G), is the graph whose vertex set is G, and two distinct elements x, y are adjacent if xy = yx. The enhanced power graph of a group G is denoted by P e (G) and is defined as a simple graph with vertex set consisting of all elements of G, where two distinct vertices x, y are adjacent if and only if x, y is a cyclic subgroup of G. The commuting graph was introduced by Brauer and Fowler in their seminal paper [21] establishing that only finitely many groups of even order can have a prescribed centraliser. The commuting graph is the complement of the non-commuting graph, first considered by Paul Erdös (cf. [65] and [40,79]). Likewise, the enhanced power graph is the complement of the noncyclic graph, introduced in [3,4]. A noncyclic graph of a group G is the graph, where two vertices x and y are adjacent if x, y is noncyclic. For technical reasons, the papers [3,4] excluded isolated vertices from consideration. Several papers also call the enhanced power graph using the terms cyclic graph or the deleted enhanced power graph, if the vertex corresponding to the identity of the group is deleted (cf. [31,32,61]).
Clearly, for any group G, we have E(P(G)) ⊆ E(P e (G)) ⊆ E(C(G)). As a concrete example, we display the power graph and the enhanced power graph of abelian group Z 2 ×Z 6 in Fig. 1. Note that C(Z 2 × Z 6 ) is a complete graph.
Many researchers have contributed to the understanding of enhanced power graphs of groups, especially after the publication of [1], and many interesting new theorems have appeared in the literature recently. The present article is a survey of the results on enhanced power graphs of finite groups. Various questions, open problems and suggestions for future work are also included. This is the first survey devoted to the enhanced power graphs of finite groups.
www.ejgta.org A survey on enhanced power graphs of finite groups | Xuanlong Ma et al.

Outline of the paper
This survey is divided into the following sections. Section 2 presents the results on the groups with dominatable enhanced power graphs, which derive from an open problem in [15]. Section 3 describes the metric dimension and strong metric dimension of an enhanced power graph. The exact values of these dimensions are computed for the enhanced power graphs of abelian groups, dihedral, semidihedral groups, and generalized quaternion groups. Section 4 deals with the perfect enhanced power graphs of groups. It gives a complete characterization of nilpotent groups that have perfect enhanced power graphs and also gives some groups whose enhanced power graphs are perfect. Section 5 is devoted to the forbidden subgraphs of enhanced power graphs. It presents a classification of all finite groups whose enhanced power graphs are split and threshold, and also classifies all finite nilpotent groups whose enhanced power graphs are chordal graphs and cographs. Several families of non-nilpotent groups whose enhanced power graphs are chordal graphs and cographs are also included. Section 6 contains the results on enhanced power graphs and other related graphs (such as power graphs and commuting graphs). Section 7 presents relevant results related to miscellaneous properties of the enhanced power graphs, including vertex connectivity, diameter, connectedness, clique number, Eulerian graphs, and so on. Section 8 deals with open problems, questions and conjectures on enhanced power graphs. The paper concludes with a brief overview of other related work supplementing the presented results.

Terminology and notation
A graph is a pair (V, E), where V is a set with elements called vertices and E is a set of edges, i.e., unordered pairs of vertices. As usual, for a graph Γ, we use V (Γ) and E(Γ) to denote its vertex and edge sets, respectively. The cardinality |V (Γ)| is called the order of Γ. The degree of u in Γ, denoted by d Γ (u) = d(u), is defined as the number of edges of Γ incident to u. A graph is simple if it has no loops or parallel edges.
Likewise, a digraph can be defined by specifying a set of vertices and a collection of ordered pairs of vertices. Each ordered pair (u, v) in the collection is called an arc (or a directed edge) from u to v. The underlying graph of a digraph is the simple undirected graph obtained by replacing each arc by an edge with the same end-vertices.
All groups considered in this survey are finite. Throughout the paper, unless stated otherwise, the word 'graph' means a finite undirected graph without loops and multiple edges. The distance between two vertices u and v in a graph Γ, denoted by d Γ (u, v) = d(u, v), is defined as the length of a shortest path between them in Γ. Note that d Γ (u, v) = 0 if u = v. The diameter of Γ, denoted by diam(Γ), is the maximum distance between two vertices of G. The strong product Γ ∆ of two graphs Γ and ∆ is the graph with vertex set V (Γ) × V (∆) in which (u, v) and (x, y) are adjacent if and only if one of u = x and vy ∈ E(H), v = y and ux ∈ E(G), and ux ∈ E(G) and vy ∈ E(H) holds. An automorphism of a graph Γ is a permutation of V (Γ) which maps adjacent vertices to adjacent vertices and nonadjacent vertices to nonadjacent vertices. The automorphism group of Γ, denoted by Aut(Γ), is the group of automorphisms of Γ under the usual composition of permutations. The vertex connectivity of a graph Γ, denoted by κ(Γ), is the minimum number of vertices which need to be removed from the vertex set of Γ so that the induced subgraph of Γ on the remaining vertices is disconnected.
Let Γ and ∆ be graphs. The graph Γ is said to be ∆-free if Γ has no induced subgraph isomorphic to ∆. A graph is a star if it is a tree with n vertices in which one vertex has degree n − 1 and every other vertex has degree 1. Denote by P n , C n , and K n the path with n vertices, the cycle of length n, and the complete graph of order n, respectively. The symbol 2K 2 denotes the bipartite graph with four vertices and two disjoint edges.
An edge colouring or edge labeling of Γ is an assignment of some colours or labels to the edges of Γ. An edge colouring of a graph is called a rainbow colouring if every pair of distinct vertices of the graph is connected by at least one path with no two edges sharing the same colour. The rainbow connection number of Γ, denoted by rc(Γ), is the minimum positive integer k for which there exists a rainbow colouring with k colours in Γ. A vertex of a graph Γ is called a dominating vertex if this vertex is adjacent to every other vertex of Γ. The vertex set of Γ is denoted by V (Γ), and the edge set is E(Γ). The distance between two vertices x and y in Γ, denoted d Γ (x, y), is the length of a shortest path from x to y. If the situation is unambiguous, we denote d Γ (x, y) simply by d(x, y). The greatest distance between any two vertices in Γ is called the diameter of Γ.
A subset of V (Γ) is called a clique if any two distinct vertices in this subset are adjacent in Γ. The clique number of Γ, denoted by ω(Γ), is the maximum cardinality of a clique in Γ. The chromatic number of a graph Γ, denoted by χ(Γ), is the smallest number of colours for V (Γ) so that adjacent vertices are coloured differently. A graph is perfect if every induced subgraph has clique number equal to chromatic number. A graph Γ is called weakly perfect if χ(Γ) = ω(Γ).
We always use G to denote a finite group, and let e denote its identity element. The order o(g) of an element g of G is the cardinality of the cyclic subgroup g . The set of orders of elements of G is denoted by π e (G). A maximal cyclic subgroup of G is a cyclic subgroup, which is not a proper subgroup of any other cyclic subgroup of G. Denote by M G the set of all maximal cyclic subgroups of G. Note that |M G | = 1 if and only if G is cyclic. The cyclic group of order n is denoted by Z n . A group G is said to be nilpotent if G has an upper central series that terminates with G. Equivalently, its central series is of finite length or its lower central series terminates with {e}. It is well known that all finite p-groups and abelian groups are nilpotent. It is well known that each finite nilpotent group is isomorphic to the direct product of its Sylow p-subgroups.
The alternating group of degree n, denoted by A n , is the group of even permutations of a set of n elements. For n ≥ 3, the dihedral group of order 2n is denoted by D 2n . Recall that it is defined by the presentation D 2n = a, b : a n = b 2 = e, bab = a −1 .
Note that D 2n = a ∪ {b, ab, a 2 b, . . . , a n−1 b}, o(a i b) = 2 for any 1 ≤ i ≤ n, and For m ≥ 2, the generalized quaternion group Q 4m of order 4m is given by For any i ≤ 1 ≤ m and any x,

Dominatable enhanced power graphs
The concept of domination is very important in graph theory. Recently, it has been considered, for example, in [13,18,37,42,45,58,68]. It is obvious that the identity element e is a dominating vertex of every enhanced power graph P e (G). Bera and Bhuniya [15] defined a dominatable enhanced power graph as an enhanced power graph with a dominating vertex other than e. For example, P e (Z 2 × Z 6 ) is a dominatable, since (0, 2) is a dominating vertex (see Fig. 1).
In [15], the authors characterized all abelian groups and all p-groups G such that P e (G) is dominatable. Ma and She [59] gave a characterization of dominatable enhanced power graphs using the following concepts. For x ∈ G, the cyclicizer [70] of x in G, denoted by Cyc(x), is defined by Since K(G) is a normal subgroup of G, the above theorem yields the following result. In [59], the authors characterized all finite nilpotent groups with dominatable enhanced power graphs. This result extends [15, Theorems 3.2 and 3.3].
where n ≥ 1, p is a prime, and H and K are nilpotent groups with 2 |H| and p |K|.
Recently, Mahmoudifar & Babai [63] gave another characterization for finite groups with dominatable enhanced power graphs. The above result implies that if G is a group with |Z(G)| = 1, then P e (G) is not dominatable. It was noted in [63] that if G is a finite group such that all its proper subgroups are dominatable, then P e (G) is not necessarily dominatable. Also, if there exists a nontrivial normal subgroup N of G such that both N and the quotient group G/N are dominatable, then P e (G) is not necessarily dominatable. Moreover, P e (G) is dominatable if G is isomorphic to the special linear group SL(2, q), where q ≥ 3 is a natural power of an odd prime. (d) A Frobenius group, then P e (G) is not dominatable.

Metric dimension and strong metric dimension of enhanced power graphs
Metric dimension was introduced by Harary and Melter [44] and, independently, by Slater [74]. It is defined as follows. A vertex z in a graph Γ is said to resolve a pair of distinct vertices x and y if d(x, z) = d(y, z). A resolving set of Γ is a subset S of V (Γ) such that every pair of distinct vertices of Γ is resolved by some vertex in S. The metric dimension of Γ, denoted by dim(Γ), is the minimum cardinality of a resolving set of Γ. It was considered, for example, in [6].
Strong metric dimension of a graph was introduced by Sebő and Tannier [72], in relation to the applications of strong resolving sets to combinatorial searching. A vertex z of a graph Γ is said to strongly resolve vertices x and y in Γ, if there exists a shortest path from z to x containing y, or a shortest path from z to y containing x. A subset S of V (Γ) is a strong resolving set of Γ if every pair of vertices of Γ is strongly resolved by some vertex in S. The smallest cardinality of a strong resolving set of Γ is denoted by sdim(Γ) and is called the strong metric dimension of Γ. The problem of computing strong metric dimension is NP-hard [56].
Ma and She [59] gave an explicit formula for the metric dimension of the enhanced power graph of a group, using the following definition. For . Note that ≡ is an equivalence relation. Denote by x the ≡-class containing the vertex x ∈ G, and let G = {x : x ∈ G}. As a corollary of Theorem 3.1, the metric dimensions of the enhanced power graphs of an elementary abelian p-group, a dihedral group, and a generalized quaternion group have been determined in [59].
Ma and Zhai [62] characterized the strong metric dimension of the enhanced power graph of a finite group. For x, y ∈ G, denote by ≈ the equivalence relation defined by It is easy to see that ≈ is an equivalence relation over G. The ≈-class containing the element x ∈ G is denoted by x. For a subset S of G and an element g ∈ G, let Panda et al. [69] and Dalal & Kumar [34] computed sdim(P e (G)) for some special groups G. (ii) If G is a noncyclic P-group of order n, then sdim(P e (G)) = n − 2; (iii) sdim(P e (D 2n )) = 2n − 2; (iv) sdim(P e (Q 4n )) = 4n − 2; (v) Let G be a noncyclic abelian p-group with order n and exponent p m . Then sdim(P e (G)) = n − m − 1.
For n ≥ 1, the group U 6n of order 6n is defined as the group generated by the elements a and b such that a 2n = b 3 = e, ba = ab −1 , that is

Perfect enhanced power graphs
It was proved in [43] that every power graph of a group is perfect. However, there exists a finite nilpotent group whose enhanced power graph is not perfect. For example, let  a 1 b 1 , b 1 c 2 , a 2 , b 2 , and a 1 c 1 form a pentagon as an induced subgraph of P e (G). It follows that P e (G) is not perfect, since a pentagon is not perfect.
Zahirović et al. [80] gave a characterization of finite nilpotent groups with perfect enhanced power graphs. On the other hand, the following theorem implies that there are many groups which do not have perfect enhanced power graphs, but whose clique number and chromatic number are equal.   Let G be a finite group, and let p 1 , p 2 , . . . , p n be all prime divisors of |G| for some n ≥ 2. If, for each i > 2, G has the unique Sylow p i -subgroup, which is cyclic, then G has perfect enhanced power graph.
The above result implies the following theorem.  The condition of Theorem 4.4 cannot be weakened by removing the requirement for the Sylow subgroups to be cyclic, nor by removing the demand for the Sylow subgroups to be unique. For example, the group Z 30 × Z 30 is abelian, and its Sylow subgroups are unique. However, P e (Z 30 × Z 30 ) is not a perfect graph (see [20]).

Forbidden subgraphs in the enhanced power graph
This section contains the results on the forbidden subgraphs of enhanced power graphs. Chordal graphs, threshold graphs, cographs and split graphs form important classes that can be defined in terms of forbidden induced subgraphs.
A graph is said to be chordal if it contains no induced cycles of length greater than 3. In other words, a chordal graph is a graph in which every cycle of length at least 4 has a chord. This means that if a chordal graph has an induced cycle C, then C is isomorphic to C 3 . A graph is called a threshold graph if it has no induced subgraph isomorphic to the path P 4 , the complete graph K 4 , or 2K 2 . Chordal graphs and threshold graphs were considered, for example, in [41] and [29]. A graph is called a cograph if this graph has no induced subgraph isomorphic to the path P 4 . A graph is said to be split if its vertex set is the disjoint union of two subsets A and B such that A induces a complete graph and B induces an empty graph.  A group is called a P -group if every nonidentity element of the group has prime order. For example, D 2q is a P -group for some odd prime q. A group is called a CP-group if every nontrivial element of the group has prime power order. For example, any p-group is a CP -group, and D 2q n is a CP -group for some odd prime q and positive integer n. Clearly, a P -group is also a CPgroup. The prime graph of a finite group G is a simple graph whose vertex set consists of all prime divisors of |G|, and two distinct vertices p and q are adjacent if there exists an element of order pq in G. The prime graph of a group was first introduced by Gruenberg and Kegel in an unpublished manuscript studying integral representations of groups in 1975.    ) Let G be a group and let p, q be two distinct primes. Then P e (G) is a chordal graph as well as a cograph if one of the following holds: (a) π e (G) = {1, p, q, pq} and G has a unique subgroup of order p or q; (b) π e (G) = {1, p, q, pq} and either G has a unique cyclic subgroup of order pq, or the intersection of all cyclic subgroups of order pq has size p or q; (c) π e (G) = {1, p, q, pq, p 2 } and either G has a unique cyclic subgroup of order pq, or the intersection of all cyclic subgroups of order pq is a where a ∈ K(G) and o(a) ∈ {p, q}. If G is a minimum order group such that P e (G) is not a chordal graph, then |G| = 36. In particular, P e (Z 6 × Z 6 ) is not chordal.

Enhanced power graph and other graphs
In this section, we collect some results on the relationships between enhanced power graph and other graphs. It was showed in [1] that a group G with these properties is one of the following: a p-group; a Frobenius group whose kernel is a p-group and complement a q-group; a 2-Frobenius group where F 1 and G/F 2 are p-groups and F 2 /F 1 is a q-group; or G has a normal 2-subgroup with quotient group H, where S ≤ H ≤ Aut(S) and S ∼ = A 5 or A 6 . All these types of group exist.
Clearly, if the vertices x and y are joined in the power graph of G, then they are joined in the commuting graph. Therefore, the power graph is a spanning subgraph of the commuting graph. It was showed in [1] that a group satisfying these conditions is either a cyclic p-group for some prime p, or satisfies the following: if O(G) denotes the largest normal subgroup of G of odd order, then O(G) is metacyclic, H = G/O(G) is a group with a unique involution, say z, and H/ z is a cyclic or dihedral 2-group, a subgroup of PΓL(2, q) containing PSL(2, q) for q an odd prime power, or A 7 . An example for the second case is the direct product of the Frobenius group of order 253 and SL (2,5). Let P * (G) be the graph obtained by deleting only the identity element of G in P(G) and this is called deleted power graph of G. Similarly, P * e (G) denotes the graph obtained by deleting only the identity element of G in P e (G) and this is called deleted enhanced power graph of G.

Miscellaneous properties of the enhanced power graph
In this section, we present some miscellaneous properties of the enhanced power graph of a group, including vertex connectivity, diameter, automorphism group, independence number, Eulerian, matching number and so on.   Theorem 1.7]) Let G be a noncyclic abelian non-p-group such that G ∼ = G 1 × Z n , gcd(|G 1 |, n) = 1 and G 1 is a p-group with no cyclic Sylow subgroup. Then κ(P e (G)) = n. For n ≥ 3, κ(P e (D 2n )) = 1, κ(P e (Q 2 n )) = 2, and κ(P e (S n )) = 1 if and only if either n or n − 1 is prime. For n ≥ 7, κ(P e (A n )) = 1 if and only if one of n, n − 1, n − 2, n/2, (n − 1)/2, (n − 2)/2 is prime.
Recently, Costanzo et al. [31] studied the deleted enhanced power graph of a direct product. For a positive integer n, let π(n) denote the set of prime divisors of n.
Theorem 7.10. ([32, Theorem D]) Let G be a group, g ∈ G, and π = π(o(g)). Write where each g p is a p-element for p ∈ π and g p g q = g q g p for all p, q ∈ π. Then g is a dominating vertex for P * e (G) if and only if, for each p ∈ π, a Sylow p-subgroup P of G is cyclic or generalized quaternion and g p ≤ P ∩ Z(G).
The above result offered a generalization of Theorem 3.2 in [15] (see Theorem 2.2). That is, for a nilpotent group G, the graph P * e (G) is dominatable if and only if G has a cyclic or generalized quaternion Sylow subgroup.  In [80], the authors defined graph S p (a), where p is a prime and a is a tuple consisting of some positive integers. where the definition of an awning can be found in [38,Definition 2.11].
An independent set of Γ is a set of vertices none of two are adjacent in Γ. The independence number of Γ, denoted by α(Γ), is the largest cardinality of an independent set of Γ. A matching of Γ is a set of edges such that no two of them are incident to the same vertex. The matching number of Γ, denoted by β(Γ), is the maximum cardinality of a matching. Theorem 7.21. ([69,Theorem 3.5]) For any finite group G. If G has odd order, then β(P e (G)) = (|G| − 1)/2. If G has even order, then where t is the number of all involutions of G. where t is the number of all involutions of G.
A group G is a 2-Frobenius group if this group has two normal subgroups K and L such that L is a Frobenius group with Frobenius kernel K and G/K is a Frobenius group with Frobenius kernel L/K. The best known example of a 2-Frobenius group is the symmetric group S 4 of order 24. Let G be a 2-Frobenius group, and let p be a prime. Assume that K is a p-group for some prime p and that G/L is not a p-group, where K and L are as in the definition. Then P * e (G) has |K| + |L/K| + m * p (G) connected components.

Open questions and other work
This section presents open problems on enhanced power graphs of groups. Note that the enhanced power graph of a group is a union of complete subgraphs on the maximal cyclic subgroups of this group. Cameron [23] put forward the following question.

Problem 1. ([23, Question 2])
Is there a simple algorithm for constructing the directed power graph or the enhanced power graph from the power graph, or the directed power graph from the enhanced power graph?
The deep commuting graph of a group was defined by Cameron and Kuzma [25]. In fact, Theorems 2.4 and 2.7 are two solutions to Problem 3. Maybe, by other methods, one can characterize all finite groups G such that P e (G) is dominatable.
where ϕ is Euler's function. Note that the bound of (2) holds for abelian groups. In fact, if G is an abelian group, then S is the set of divisors of the exponent of G, and so the sum on the right of (2) is the exponent of G, which is the largest element order in G. If a finite graph Γ is isomorphic to an induced subgraph of the enhanced power graph of some finite group G, then we say that Γ is embeddable in the enhanced power graph of G. Bošnjak et al. [20] and Zahirović et al. [80] studied perfect enhanced power graphs of groups. Ma, Lv, and She [60] classified finite groups whose enhanced power graphs are split graphs and threshold graphs. They also classified finite nilpotent groups whose enhanced power graphs are chordal graphs and cographs, and gave some families of non-nilpotent groups whose enhanced power graphs are chordal graphs and cographs. These results provide a partial solution to Problem 7. , then we may collapse them to a single vertex; this process is called twin reduction. Clear, the automorphism group Aut(Γ) of a graph Γ preserves twin relations on Γ, and that vertices in a twin equivalence class can be permuted arbitrarily. It follows by induction that the group induces an automorphism group on the cokernel Γ * of Γ, say Aut − (Γ * ). The twin reduction on Γ is faithful if Aut − (Γ * ) = Aut(Γ * ). In [1], the authors solved Problems 11 and 12 for the special case of finite groups (see [1], Theorems 28 and 30, respectively).

Problem 13. ([1, Question 31])
What can be said about the difference of the enhanced power graph and the power graph, or the difference of the commuting graph and the enhanced power graph? In particular, for which groups is either of these graphs connected?
The multiset dimension was introduced by Rinovia Simanjuntak et al. [73] as a variation of metric dimension. It was considered in [7,19]. Here we suggest the following open problem.
Problem 14. For any finite group G, describe the multiset dimension of the enhanced power graph of G.
The locating chromatic number of a graph was introduced by Chartrand et al. [28]. It is equal to the minimum number of colours needed in a locating colouring of the graph. Let us refer to [12] for more information on the locating-chromatic number. We propose the following open problem.
Problem 15. For any finite group G, describe the locating-chromatic number of the enhanced power graph of G.
Problem 16. Investigate enhanced power graphs with labelings of various essential classes. For any finite group G, describe α-labelings, cordial labelings, distance labelings, face-antimagic labelings, graceful labelings, integer additive set-labelings, irregular total labelings, magic and antimagic graph labelings, super antimagic labelings, supermagic labelings, and vertex irregular reflexive labelings of the enhanced power graph of G.
In conclusion, we mention a few pertinent references which complement the results presented above and can also be used in the future work. The proper enhanced power graph of a group G is the induced subgraph of the enhanced power graph on the set G \ D, where D is the set of dominating vertices of the enhanced power graph of G. Bera and Dey [16] classified all nilpotent groups such that their proper enhanced power graphs are connected and determined their diameters. They also explicitly found the domination number of the proper enhanced power graph of a finite nilpotent group. The enhanced power graph or cyclic graph of a semigroup S is a simple graph whose vertex set is S and two vertices x, y ∈ S are adjacent if and only if x, y ∈ z for some z ∈ S, where z is the subsemigroup of S generated by z. In [33,35,36,5], several graph theoretical properties of the enhanced power graph of semigroups were studied, including the dominating number, independence number, genus, connectedness, minimum degree, chromatic number and so on. Moreover, the authors characterized the semigroups S such that the enhanced power graph of S is complete, bipartite, regular, is a tree, or is a null graph. Moreover, the structure of the enhanced power graph of any semigroup was described. Dupont et al. [39] introduced the enhanced quotient graph of the quotient of a finite group, and gave necessary and sufficient conditions for the graph to be complete and Eulerian.