On the connectivity of enhanced power graph of finite group
aa r X i v : . [ m a t h . C O ] J u l ON THE CONNECTIVITY OF ENHANCED POWER GRAPH OFFINITE GROUP
SUDIP BERA, HIRANYA KISHORE DEY, AND SAJAL KUMAR MUKHERJEE
Abstract.
This paper deals with the vertex connectivity of enhanced power graph of finitegroup. We classify all abelian groups G such that vertex connectivity of enhanced powergraph of G is 1 . We derive an upper bound of vertex connectivity for the enhanced powergraph of any general abelian group G. Also we completely characterize all abelian group G, such that the proper enhanced power graph is connected. Moreover, we study some specialclass of non-abelian group G such that the proper enhanced power graph is connected andwe find their vertex connectivity. Introduction
The exploration of graphs associated with algebraic structures is important, as graphslike these enrich both algebra and graph theory. Besides, they have important applications(see, for example, [2, 21]) and are related to automata theory [22]. During the last twodecades, investigation of the interplay between the properties of an algebraic structure S and the graph-theoretic properties of Γ( S ) , a graph associated with S, has been an excitingtopic of research. Different types of graphs, specifically zero-divisor graph of a ring [3],semiring [5], semigroup [14], poset [20], power graph of semigroup [12, 24], group [23], normalsubgroup based power graph of group [8], intersection power graph of group [6] etc. havebeen introduced to study algebraic structures using graph theory. Chakrabarty et. al in [12]introduced the undirected power graph of a semigroup in the following way. Definition 1.1 ([12]) . Let S be a semigroup, then the power graph P ( S ) of S, is a simplegraph, whose vertex set is S and two distinct vertices u and v are edge connected if and onlyif either u m = v or v n = u, where m, n ∈ N . Another well-studied graph, called commuting graph associated with a group G is studiedin [10] as a part of the classification of finite simple groups. For more information about thecommuting graph, see [4, 17]. Definition 1.2 ([10]) . Let G be a group, then the commuting graph of G, denoted by C ( G ) , is the simple graph whose vertex set is non-central elements of G and two distinct vertices u and v are adjacent if and only if uv = vu. Definition 1.3 ([1]) . Given a group G, the enhanced power graph of G, denoted by G e ( G ) , is the graph with vertex set G, in which u and v are joined if and only if there exists anelement w ∈ G such that both u and v are powers of w. Mathematics Subject Classification.
Key words and phrases. abelian group; dominating vertex; enhanced power graph; vertex connectivity.
The authors in [1] measure how close the power graph is to the commuting graph byusing the enhanced power graph. In fact, the enhanced power graph contains the powergraph and is a subgraph of the commuting graph. They characterized the finite groupssuch that, for an arbitrary pair of these three graphs for which this pair of graphs areequal. Besides, in [29], the researchers proved that finite groups with isomorphic enhancedpower graphs have isomorphic directed power graphs. They showed that any isomorphismbetween the undirected power graph of finite groups is an isomorphism between enhancedpower graphs of these group. Ma and She in [25] derived the metric dimension of enhancedpower graphs of finite groups where as Hamzeh et.al in [19] derived the automorphismgroups of enhanced power graphs of finite groups. Recently Panda et.al in [26] have studiedindependence number, vertex covering number and some other graph invariants of enhancedpower graphs.1.1.
Basic definitions, Notations and Main Results.
For the convenience of the readerand also for later use, we recall some basic definitions and notations about graphs. LetΓ = (
V, E ) be a graph where V is the set of vertices and E is the set of edges. Two elements u and v are said to be adjacent if ( u, v ) ∈ E. The standard distance between two vertices u and v in a connected graph Γ is denoted by d ( u, v ) . Clearly, if u and v are adjacent, then d ( u, v ) = 1 . For a graph Γ , its diameter is defined as diam (Γ) = max u,v ∈ V d ( u, v ) . That is,the diameter of graph is the largest possible distance between pair of vertices of a graph. A path of length l between two vertices v and v k is an alternating sequence of vertices andedges v , e , v , e , v , · · · , v k − , e k − , v k , where the v ′ i s are distinct (except possibly the firstand last vertices) and e i is the edge ( v i , v i +1 ) . A graph Γ is said to be connected if for anypair of vertices u and v, there exists a path between u and v. Γ is said to be complete if anytwo distinct vertices are adjacent. A vertex of a graph Γ = (
V, E ) is called a dominatingvertex if it is adjacent to every other vertex. For a graph Γ , let Dom (Γ) denote the set ofall dominating vertices in Γ . The vertex connectivity of a graph Γ , denoted by κ (Γ) is theminimum number of vertices which need to be removed from the vertex set Γ so that theinduced subgraph of Γ on the remaining vertices is disconnected. The complete graph with n vertices has connectivity n − . For more on graph theory we refer [9, 18, 28]. The enhancedpower graph is called dominatable if it has a dominating vertex other than identity.Throughout this paper we consider G as a finite group. | G | denotes the cardinality of theset G. For a prime p, a group G is said to be a p -group if | G | = p r , r ∈ N . For any element g ∈ G, o( g ) denotes the order of the element g ∈ G. Let G be a group and a ∈ G, then G en ( a ) is the set of all generators of the cyclic group h a i . Let m and n be any two positiveintegers, then the greatest common divisor of m and n is denoted by gcd( m, n ) . The Euler’sphi function φ ( n ) is the number of integers k in the range 1 ≤ k ≤ n for which the gcd( n, k )is equal to 1 . The set { , , · · · , n } is denoted by [ n ] . In this paper, our focus is on the vertex connectivity of enhanced power graphs of finiteabelian groups. If G is a non-cyclic non-generalized quaternion p -group, then we determinethe exact value of the vertex connectivity of G e ( G ) . Theorem 1.1.
Let G be a finite p -group such that G is neither cyclic nor generalized quater-nion group. Then κ ( G e ( G )) = 1 . Our next result classifies all non-cyclic abelian groups G such that κ ( G e ( G )) = 1 . N THE CONNECTIVITY OF ENHANCED POWER GRAPH OF FINITE GROUP 3
Theorem 1.2.
Let G be a finite non-cyclic abelian group. Then κ ( G e ( G )) is if and onlyif G is a p -group. The authors in [1, Question 40] asked about the connectivity of power graphs when allthe dominating vertices are removed. Recently, Cameron and Jafari in [11] answered thisquestion for power graphs. In this paper, we investigate the same question for enhancedpower graphs. To seek the answer of this question, define the following graph:
Definition 1.4.
Given a group G, the proper enhanced power graph of G, denoted by G ∗∗ e ( G ) , is the graph obtained by deleting all the dominating vertices from the enhanced power graph G e ( G ) . Moreover, by G ∗ e ( G ) we denote the graph obtained by deleting only the identityelement of G and this is called deleted enhanced power graph of G. Note that if there is nosuch dominating vertex other than identity, then G ∗ e ( G ) = G ∗∗ e ( G ) . In [7], Bera et.al characterized all abelian groups G, such that | Dom( G e ( G )) | > . In fact,they proved the folllowing:
Theorem 1.3 ([7]) . Let G be a finite abelian group. Then G e ( G ) is dominatable if and onlyif G has a cyclic Sylow subgroup. So, from Theorem 1.3, | Dom( G e ( G )) | > G = G × Z n , where gcd( | G | , n ) = 1and G has no cyclic sylow subgroup. Then one natural question is that which are thedominatable vertices of G e ( G ) . The next theorem gives the complete list of the dominatingvertices of G e ( G ) . Theorem 1.4.
Let G be a non-cyclic abelian group such that G has no cyclicsylow subgroup. If n ∈ N , and gcd ( | G | , n ) = 1 , then Dom ( G e ( G × Z n )) = { ( e, x ) , where x is any element of Z n and e is the identity of G } . Theorem 1.2 completely characterizes the connectivity of G ∗ e ( G ) for any finite abelian p -group G. Now if G is a non-cyclic abelian non p -group such that G has no cyclic sylowsubgroup, then by Theorem 1.3, G has no dominating vertex other than the identity. So,in this case we care about the connectivity of G ∗∗ e ( G ) = G ∗ e ( G ) and by Theorem 1.2, G ∗ e ( G )is connected. Therefore when the graph G ∗ e ( G ) has a dominating vertex other than ientity,the connectivity of G e ( G ) is a more interesting question. In this paper, we characterize forwhich finite abelian groups, the proper enhanced power graphs G ∗∗ e ( G ) are connected andfor which they are not. Our contributions on this paper in this theme is the following: Theorem 1.5.
Let G be a non-cyclic abelian non p -group such that G ∼ = G × Z n , gcd ( | G | , n ) = 1 and G has no cylcic sylow subgroup. Then G ∗∗ e ( G ) is disconnectedif and only if G is a p -group. Therefore, from Theorem 1.5, when G is not a p -group, G ∗∗ e ( G ) remains connected. Thus,the number of additional vertices required to make it disconnected is an interesting question.On this theme, our next result is the following: Theorem 1.6.
Let G be a non-cyclic abelian group such that G ∼ = Z p t × Z p t × Z p t ×· · ·× Z p t k × Z p t × Z p t ×· · ·× Z p t k ×· · ·× Z p tr r × Z p tr r ×· · ·× Z p trkrr , SUDIP BERA, HIRANYA KISHORE DEY, AND SAJAL KUMAR MUKHERJEE where k i ≥ and ≤ t i ≤ t i ≤ · · · ≤ t ik i , for all i ∈ [ r ] . Then κ ( G e ( G )) ≤ p t p t · · · p t r r − φ ( p t p t · · · p t r r ) . When G is a p -group, the following result gives the exact value of the vertex connectivityof G e ( G ) . Theorem 1.7.
Let G be a non-cyclic abelian non- p -group such that G ∼ = G × Z n , gcd ( | G | , n ) = 1 and G is a p -group with no cyclic sylow subgroup. Then κ ( G e ( G )) = n. Throughout this paper, the group operation of any abelian group is taken to be additive.2.
Preliminaries
We first recall some earlier known results on enhanced power graphs which we will needthroughout the paper. In [7], Bera et.al studied about the completeness, dominatability andmany other properties of enhanced power graph of finite group. In fact, they proved thefollowing:
Lemma 2.1 (Theorem 2.4, [7]) . The enhanced power graph G e ( G ) of the group G is completeif and only if G is cyclic. Lemma 2.2 (Theorem 3.3, [7]) . Let G be a non-abelian -group. Then the enhanced powergraph G e ( G ) is dominatable if and only if G is generalized quarternion group. Lemma 2.3 (Theorem 3.1, [7]) . Let G be a finite group and n ∈ N . If gcd ( | G | , n ) = 1 , thenthe enhanced power graph G e ( G × Z n ) is dominatable. Lemma 2.4 (Lemma 2.1, [7]) . Let a, b ∈ G with o ( a ) = o ( b ) and h a i 6 = h b i . Then x is notadjacent with y for every x ∈ G en ( a ) and y ∈ G en ( b ) . We next prove some important lemmas which are used to prove our main theorems.
Lemma 2.5.
Let G be a finite group and x, y ∈ G \ { e } be such that gcd ( o ( x ) , o ( y )) = 1 and xy = yx. Then, x ∼ y in G ∗ e ( G ) .Proof. Let o( a ) = m and o( b ) = n. Now gcd( m, n ) = 1 implies that n φ ( m ) = mk + 1 , k ∈ N , ( by Euler’s formula ) . Again, ab = ba implies that ( ab ) n φ ( m ) = a n φ ( m ) b n φ ( m ) = a n φ ( m ) = a mk +1 = a. As a result, a ∈ h ab i . Similarly we can prove that b ∈ h ab i . Consequently, a ∼ b in G ∗ ( G ) . (cid:3) Lemma 2.6.
Let G be a p -group. Let a, b be two elements of G of order p, p i ( i ≥ respectively. If there is a path between a and b in G ∗ e ( G ) , then h a i ⊂ h b i . In particular, ifboth a and b have order p, then, h a i = h b i . Proof.
Let a = a ∼ a ∼ · · · ∼ a m = b be a path between a and b. Now a = a ∼ a impliesthat there exists x ∈ G such that a , a ∈ h x i . As a result, a ∈ h a i (since a cyclic grouphas a unique subgroup corresponding to each divisor of the order of the cyclic group). Now, a ∼ a and G is a p -group, then either a ∈ h a i or a ∈ h a i . Clearly for both of the cases a ∈ h a i . Continuing this process we can conclude that a ∈ h b i . (cid:3) Lemma 2.7.
Let G be any non-cyclic group. For any dominating vertex v ( = e ) of G thereexists a prime p dividing o ( v ) such that G has a unique subgroup of order p. N THE CONNECTIVITY OF ENHANCED POWER GRAPH OF FINITE GROUP 5
Proof.
Let v = e be a dominating vertex and o( v ) = m . Let p be a prime divisor of m and m = rp. We claim that H = h v r i is the unique subgroup of order p in G. Consider x ∈ G such that o( x ) = p. Since v is dominating vertex, we have x ∼ v. Thus, there exists a cyclicsubgroup A such that x, v ∈ A. Then o( x ) = p implies that x ∈ h v i . If x = v q , by divisionalgorithm it can be shown that q has to be a multiple of r and thus x ∈ H. This completesthe proof. (cid:3)
We next move on to the most important result of this section.
Theorem 2.8.
For any group G , the graph P ∗ ( G ) is connected if and only if the graph G ∗ e ( G ) is connected.Proof. The forward implication is easy. That is, if P ∗ ( G ) is connected then G ∗ e ( G ) is ofcourse connected. We prove the other direction. Let, G ∗ e ( G ) be connected and a, b ∈ P ∗ ( G ) . As G ∗ e ( G ) is connected, there exists a path between a = a ∼ a ∼ · · · ∼ a m = b. Now, a i ∼ a i +1 in G ∗ e ( G ) = ⇒ there exist b i ∈ G such that both a i and a i +1 ∈ h b i i . In that case, a i ∼ b i ∼ a i +1 . Therefore, we have a = a ∼ b ∼ a ∼ b ∼ a · · · ∼ a m = b in P ∗ ( G ) . Thiscompletes the proof. (cid:3)
Therefore, for any graph G , the information about the connectivity of one of the twographs P ∗ ( G ) and G ∗ e ( G ) gives information about the connectivity of the other one.3. Proofs of main results about Vertex Connectivity of G e ( G ) when G isabelian Proof of Theorem 1.1.
First suppose that G is non-cyclic abelian p -group. Clearly G has atleast two distinct cyclic subgroups H = h a i and H = h b i of order p. Now by Lemma 2.6,there is no path joining a and b in G ∗ e ( G ) , otherwise H = H . The proof is complete. (cid:3)
Proof of Theorem 1.2. G is non-cyclic abelian p -group. Therefore, by Theorem 1.1, κ ( G e ( G )) = 1 . For the converse part, let G be a finite abelian group which is not a p -group. Let, p , p , · · · , p k be the prime factors of | G | . Let, a, b ∈ G and o( a ) = p r p r · · · p r k k ando( b ) = p s p s · · · p s k k . We consider the following two cases:Case 1: There exists distinct i and j with r i = 0 and s j = 0. Then the elements p r p r · · · p r i − i − p r i +1 i +1 · · · p r k k a and p s p s · · · p s j − j − p s j +1 j +1 · · · p s j j b are of order p r i i and p s j j respectively.Thus, by Lemma 2.5, p r p r · · · p r i − i − p r i +1 i +1 . . . p r k k a and p s p s · · · p s j − j − p s j +1 j +1 · · · p s j j b are adjacent.Therefore we have a ∼ p r p r · · · p r i − i − p r i +1 i +1 · · · p r k k a ∼ p s p s · · · p s j − j − p s j +1 j +1 · · · p s j j b ∼ b. That is, there exists a path of length ≤ a and b . We observe that this case takescare of everything except when both o( a ) and o( b ) are power of the same prime p ℓ for some1 ≤ ℓ ≤ k which we consider next.Case 2: o( a ) = p r ℓ ℓ and o( b ) = p s ℓ ℓ . Let, c be an element of order p i in G with i = ℓ . Then byLemma 2.5, we have a ∼ c ∼ b . Thus, G ∗ e ( G ) is connected. This completes the proof. (cid:3) SUDIP BERA, HIRANYA KISHORE DEY, AND SAJAL KUMAR MUKHERJEE
By Theorem 2.8, we immediately get the following corollary on the connectivity of thepower graph.
Corollary 1.
Let G be a finite non-cyclic abelian group. Then κ ( P ( G )) is if and only if G is a p -group.Proof of Theorem 1.4. We show that ( e, x ) is a dominating vertex, where e is the identityelement of the group G and x is a any element of the group Z n . Consider an arbitrary vertex( g, y ) of the graph G e ( G × Z n ) . Case 1: Let g = e. Let a be a generator of the cyclic group Z n . Now y ∈ Z n implies that( e, y ) , ( e, x ) ∈ h ( e, a ) i and so ( e, x ) ∼ ( e, y ) . Case 2: Let g = e and y = 0 . [Here 0 actually means the additive identity of the group Z n ] . We show that ( g, , ( e, x ) ∈ h ( g, a ) i . First we show that ( e, a ) ∈ h ( g, a ) i . Let o( g ) = m. Nowgcd( | G | , n ) = 1 implies that gcd( m, n ) = 1 . Then by Euler’s Theorem m φ ( n ) = nℓ + 1 , ℓ ∈ N . Therefore, ( g, a ) m φ ( n ) = ( g m φ ( n ) , a m φ ( n ) ) = ( e, a nℓ +1 ) = ( e, a ) . Hence, ( e, a ) ∈ h ( g, a ) i . Nowwe show that ( g, ∈ h ( g, a ) i . It is given that gcd( m, n ) = 1 . So, by the Euler’s theorem, n φ ( m ) = mk + 1 , k ∈ N . Hence ( g, a ) n φ ( m ) = ( g n φ ( m ) ,
0) = ( g mk +1 ,
0) = ( g, . Consequently,( g, ∈ h ( g, a ) i . Case 3: Let g = e and y = 0 . We show that ( g, y ) , ( e, x ) ∈ h ( g, a ) i . Already we have provedthat ( g, , ( e, x ) ∈ h ( g, a ) i . Since a is a generator of Z n , ( e, y ) ∈ h ( e, a ) i ⊂ h ( g, a ) i . Hence( g, y ) = ( g, e, y ) ∈ h ( g, a ) i . To finish the proof we have to show that if ( g, z ) is a dominating vertex, then g must bethe identity of G. Let G = Z p t × Z p t × Z p t ×· · ·× Z p t k × Z p t × Z p t ×· · ·× Z p t k ×· · ·× Z p tr r × Z p tr r ×· · ·× Z p trkrr , where k i ≥ t i ≤ t i ≤ · · · ≤ t ik i , for all i ∈ [ r ]. Let v = ( x p t , x p t , · · · x p t k , x p t , · · · , x p t k , · · · , x p t rr , x p t rr , · · · , x p trkrr , z )be a dominating vertex. We will prove that, for each i ∈ [ r ] and j ∈ [ k i ] , x p ti i = x p ti i = · · · = x p tiji = 0 . [Here 0 actually means the additive identity of the group Z p tiji ]. Considerthe element v ′ = (0 , , · · · , , g p t k , , , · · · , g p t k , , · · · , , g p trkrr , z ′ ) , where g p tikii is a generator of the cyclic group Z p tikii for each i ∈ [ r ] and z ′ is a generator of Z n . As v is a dominating vertex of the graph G e ( G ) , we have v ∼ v ′ in G e ( G ) . Clearly, v ′ is an element of maximum ordered. So, we have v ∈ h v ′ i . As a result, for each i ∈ [ r ] and j ∈ [ k i − , x p ti i = x p ti i = · · · = x p tiji = 0 , i.e., v = (0 , , · · · , , x p t k , , , · · · , , x p t k , , · · · , , x p trkrr , z ) . Now we show that x p tikii = 0 , for all i ∈ [ r ] . Suppose at least one of the x p tikii is non-zero.Without any loss of generality we assume that x p t k = 0 . Consider v = ( x, , , · · · , , , , (last zero is the identity of cyclic group Z n ) where x ∈ Z p t k with o( x ) = p . Then p divides o( v ) . If v ∼ v , then there exists a cyclic subgroup C of G × Z n such that v, v ∈ C. N THE CONNECTIVITY OF ENHANCED POWER GRAPH OF FINITE GROUP 7
Then o( v ) = p and p divides o( v ) implies that v ∈ h v i , which contradicts that x = 0 . This completes the proof. (cid:3)
We next take care about the connectivity of the proper enhanced power graph G ∗∗ e ( G ) , when G is abelian. Proof of Theorem 1.5.
First we show that if G is a p -group, then G ∗∗ e ( G ) is disconnected.By Theorem 1.4, order of each element of G ∗∗ e ( G ) is divisible by p. So applying the proofof the lemma 2.6, we get that for two elements a = ( x,
0) and b = ( x ′ , y ′ ) of G ∗∗ e ( G ) , witho(( x, p and x ′ = e, if there exists any path joining a and b, then h ( x, i is containedin or equal to h ( x ′ , y ′ ) i . In particular, if both a and b have order p, then the existence of apath joining a and b implies that h a i = h b i . Since, G is noncyclic abelian p -group, thereexist two elements a and b of order p such that h a i 6 = h b i . So by our previous observation, a is not path connected to b. Conversely, suppose that G is non- p -group. Then we show that G ∗∗ ( G ) is connected.Here we have two cases.Case 1: Let ( x ,
0) and ( x ,
0) be two elements of G × Z n such that x = e and x = e. Then by same argument as in proof of converse part of Theorem 1.2, ( x ,
0) and ( x ,
0) arepath connected in G ∗∗ e ( G ) . Case 2: Let ( x , y ) , ( x , y ) ∈ V ( G ∗∗ e ( G )) . Clearly, ( x , y ) ∼ ( x ,
0) and ( x , ∼ ( x , y ) . Again there is a path between ( x ,
0) and ( x ,
0) in G ∗∗ e ( G ) by Case 1. Therefore, ( x , y ) and( x , y ) are path connected in G ∗∗ e ( G ) . Hence the graph G ∗∗ e ( G ) is connected. This completesthe proof. (cid:3) Proof of Theorem 1.6.
Let H = h a i be a cyclic subgroup of G, where a =( a , , · · · , , a , , · · · , a r , · · · , , and a i ∈ Z p ti i such that o( a i ) = p t i i , for i =1 , , · · · , r. H is maximal cyclic subgroup of G. Now we show that for any b ∈ G \ H, there isno edge between b and any element in G en ( a ) . If possible there exists x ∈ G en ( a ) such that b ∼ x in G e ( G ) . Then there exists a cyclic subgroup K of G such that b, x ∈ K. Again H is a maximal cyclic subgroup of G which is also generated by x . Therefore, K = h a i = H. Hence a contradiction as b ∈ G \ H. Clearly, if we remove the identity and non-identitynon-generators elements from the cyclic subgroup H, then the graph will be disconnectedand the number of deleted vertices is p t p t · · · p t r r − φ ( p t p t · · · p t r r ) . Hence the result. (cid:3)
From Theorem 1.6, we immediately have the following corollary on the vertex connectivityof power graphs of any non-cyclic abelian group.
Corollary 2.
Let G be a non-cyclic abelian group such that G ∼ = Z p t × Z p t × Z p t ×· · ·× Z p t k × Z p t × Z p t ×· · ·× Z p t k ×· · ·× Z p tr r × Z p tr r ×· · ·× Z p trkrr , where k i ≥ and ≤ t i ≤ t i ≤ · · · ≤ t ik i , for all i ∈ [ r ] . Then κ ( P ( G )) ≤ p t p t · · · p t r r − φ ( p t p t · · · p t r r ) . Proof of Theorem 1.7.
Proof of this theorem follows from Theorems 1.4 and 1.5. (cid:3)
SUDIP BERA, HIRANYA KISHORE DEY, AND SAJAL KUMAR MUKHERJEE
Number of Components of G ∗∗ e ( G ) when it is disconnected. So far, we havecharacterized the abelian groups for which the proper enhanced power graph is disconnected.In this context, the natural question that comes to our mind is the number of connectedcomponents of the subgraph G ∗∗ e ( G ) . By Theorem 1.5, the proper enhanced power graph G ∗∗ e ( G ) for a finite abelian group G is disconnected when G is either a non-cyclic p -groupor G ∼ = G × Z n where G is an p -group and gcd( p, n ) = 1 . Here, we explicitly count thenumber of components for those G . Theorem 3.1.
Let G be a finite abelian p -group. Suppose G = Z p t × Z p t × · · · × Z p tr . where r ≥ and t ≤ t ≤ · · · ≤ t r . Then, the number of components of G ∗∗ e ( G ) is p r − p − . Proof.
It is easy to show that there are p r − p . For any element a oforder p , the p − a must be in the same component. Moreover,by Lemma 2.6, if any two elements of order p are connected by a path, then one of themmust be the multiple of another. Henceforth, there are exactly p − p inany component. Thus, the number of connected components of G ∗ e ( G ) is p r − p − . (cid:3) It is quite interesting to note that the number of components of the proper enhanced powergraph of a finite abelian non-cyclic p -group is independent of the exponent t i ’s. In the nextresult, we prove that this phenomenon is observed also in the case when G ∼ = G × Z n where G is an p -group and gcd( p, n ) = 1 . Let C ( G ∗∗ e ( G )) be the set of connected components ofthe proper enhanced power graph G ∗∗ e ( G ) . Theorem 3.2.
Let G be an abelian group such that G ∼ = Z p t × Z p t × · · · × Z p tr × Z n , where r ≥ and gcd ( p, n ) = 1 . Then, the number of components of G ∗∗ e ( G ) is p r − p − . Proof.
Let, G = Z p t × Z p t × · · · × Z p tr . By Theorem 3.1, the number of connectedcomponents of G ∗∗ e ( G ) is p r − p − . Let, C , C , · · · , C pr − p − be the components of G ∗∗ e ( G ) C ( G ∗∗ e ( G × Z n )) by f ( C i ) = C i × Z n . At first, we show that there is no path in between C i × Z n and C j × Z n for 1 ≤ i < j ≤ p r − p − . Let, there exists an path between ( a , b ) and ( a , b ) where a ∈ C i , a ∈ C j and b , b ∈ Z n . If possible, let ( a , b ) ∼ ( c , d ) ∼ ( c , d ) ∼ · · · ∼ ( c m − , d m − ) ∼ ( a , b ) in G ∗∗ e ( G )where c , c , . . . , c m − ∈ G and d , d , . . . , d m − ∈ Z n . Then c , c , · · · , c m − must be non-zero elements of G . This proves that a and a are connected by a path in G ∗ e ( G ) whichcontradicts the fact that C i and C j are distinct connected components of G ∗∗ e ( G ) . Therefore,the number of components of G ∗∗ e ( G ) is at least p r − p − . Moreover, it is clear that the number of elements of order p in G is p r − p − . Any elementof order > p is adjacent to an element of order p . Therefore, the number of components of G ∗∗ e ( G ) should be exactly equal to p r − p − . The proof is complete. (cid:3)
N THE CONNECTIVITY OF ENHANCED POWER GRAPH OF FINITE GROUP 9 Vertex Connectivity of Some Non-abelian groups
In this Section, we discuss the vertex connectivity of some interesting classes of non-abelian groups. We start with the dihedral groups. We need the structures of these groupsto determine the vertex connectivity. For n ≥
2, the dihedral group of order 2 n is definedby the following presentation: D n = h r, s : r n = s = e, rs = sr − i . We also consider the generalized quarternion groups Q n . Let x = (1 ,
0) and y = (0 , . Then Q n = h x, y i , where(1) x has order 2 n − and y has order 4 , (2) every element of Q n can be written in the form x a or x a y for some a ∈ Z , (3) x n − = y , (4) for each g ∈ Q n such that g ∈ h x i , such that gxg − = x − . For more information about D n , and Q n see [13, 16, 27]. Theorem 4.1.
Let G be the dihedral group of order n. Then κ ( G e ( G )) is . Moreover, thenumber of components of G ∗∗ e ( G ) is n + 1 .Proof. Consider the following n + 1 sets: S = { rs } , S = { r s } , · · · , S n − = { r n − s } , S n = { s } , S n +1 = { r, r , . . . , r n − } . We observe that G \ { e } = ∪ n +1 i =1 S i and for 1 ≤ i < j ≤ n . Moreover, the power of anyelement of S i must be in S i itself in G \ { e } . Therefore, there can be no edge between S i and S j for distinct i, j . This completes the proof. (cid:3) Theorem 4.2.
For n ≥ , let Q n be the generalized quaternion group. Then the vertexconnectivity of G e ( Q n ) is . Moreover the number of components of G ∗∗ e ( Q n ) is n − + 1 . Proof. Q n is generalized quaternion group, so Q n is a 2-group and it has a unique minimalsubgroup of order 2 . Let g ∈ Q n such that o( g ) = 2 . Then by Lemma 2.2, g and e areadjacent to all other vertices in G e ( Q n ) . For this reason to disconnect the graph we have todelete the vertices e, g.
Consequently, κ ( G e ( Q n )) ≥ . Now we show that after removing thevertices e and g from V ( G e ( Q n )) , the graph G ∗∗ e ( Q n ) will be disconnected. Let x ∈ Q n suchthat o( x ) = 2 n − and H = h x i . We will prove that there is no edge between the vertices in H and Q n \ H. If for any y ∈ Q n \ H is adjacent to a vertex of H, then y should belong to H, (since H is the only subgroup of order 2 n − and there is no other subgroup of order > n − )a contradiction. So the graph is disconnected. Moreover it is clear that there is no edgebetween the vertices in G en ( y ) and G en ( z ) , where y, z ∈ Q n \ H such that o( y ) = 4 = o( z )and h y i 6 = h z i . Hence the number of components in G ∗∗ e ( Q n ) is 2 n − + 1 (as the number of4-ordered element in Q n \ H is 2 n − and φ (4) = 2) . (cid:3) Corollary 3.
Let Q n be the generalized quaternion group. Then the enhanced power graph G ∗ e ( Q n ) is connected but the proper enhanced power graph G ∗∗ e ( Q n ) is disconnected. We next consider the family of symmetric groups S n . [16] is a good reference for this.Recall from Section 1 that P ( G ) denotes the power graph of G. Let, P ∗ ( G ) denote the power graph of G after deleting the identity. Doostabadi et.al proved the following theoremon the vertex connectivity of power graphs [15]. Theorem 4.3.
Let G = S n be a symmetric group with n ≥ . Then (1) If n ≥ and neither n nor n − is a prime, then P ∗ ( G ) is connected. (2) If n is such that either n or n − is a prime, then P ∗ ( G ) is disconnected. We prove an analogous result corresponding to the enhanced power graph of S n . For this,we first prove that G e ( S n ) has no dominating vertex. Lemma 4.4.
For n ≥ , the enhanced power graph G e ( S n ) has no dominating vertex otherthan identity. Therefore, for n ≥ , the graphs G ∗ e ( S n ) and G ∗∗ e ( S n ) coincide.Proof. We first prove it when n is composite. Let a = e be a dominating vertex. By Lemma2.7, there exists a prime p dividing o( a ) such that G has a unique subgroup of order p. Butwe can take C = h (1 , , · · · , p ) i and C = h (2 , , · · · , p + 1) i and arrive at a contradiction.We next consider the case when n is prime, say n = p. Let a = e be a dominating vertex.Then, a ∼ h (1 , , · · · , p ) i . So, they are contained in a cyclic subgroup, say A of G. Now, therecannot be a subgroup of G which properly contains h (1 , , · · · , p ) i . Hence, a ∈ h (1 , , · · · , p ) i and consequently h a i = h (1 , , · · · , p ) i . Again, since a is a dominating vertex, a ∼ (1 , . Now, applying the similar argument as above, we see that (1 , ∈ h a i , which is not possible.The proof is complete. (cid:3) Theorem 4.5.
For positive integers n ≥ , κ ( G e ( S n )) = 1 if and only if either n or n − is prime.Proof. By Lemma 4.4, we have G ∗ e ( S n ) = G ∗∗ e ( S n ) . So, by Theorem 2.8, G ∗∗ e ( S n ) is connectedif and only if P ∗ ( S n ) is connected. Now, the proof is complete using Theorem 4.3. (cid:3) Let, A n ⊆ S n be the alternating group. The family of Alternating groups is an interest-ing subgroup of the set of even permutations in S n . Doostabadi et.al proved the followingtheorem on the vertex connectivity of power graphs [15].
Theorem 4.6.
Let G = A n be the alternating group and n ≥ . Then (1) If n, n − , n − , n/ , ( n − / , ( n − / are not primes, then P ∗ ( G ) is connected. (2) If n is such that any one of n, n − , n − , n/ , ( n − / , ( n − / is prime, then P ∗ ( G ) is not connected. We start with showing that G e ( A n ) has no dominating vertex. Lemma 4.7.
For n ≥ , the enhanced power graph G e ( A n ) has no dominating vertex otherthan identity. Therefore, for n ≥ , the graphs G ∗ e ( A n ) and G ∗∗ e ( A n ) coincide.Proof. Let, a = e be a dominating vertex. By Lemma 2.7, there exists a prime p suchthat G has a unique subgroup of order p. If p ≥
4, take C = h (1 , , , · · · , p ) i and C = h (1 , , , · · · , p ) i . If p = 3 , take C = h (1 , , i and C = h (2 , , i . When p = 2 , let C = h (1 , , i and C = h (1 , , i . Thus, in each case, we can verify that both C and C are even permutations and this contradicts Lemma 2.7. This completes the proof. (cid:3) From Theorem 4.6 and Lemma 4.7, we prove the following theorem.
N THE CONNECTIVITY OF ENHANCED POWER GRAPH OF FINITE GROUP 11
Theorem 4.8.
For positive integers n ≥ , κ ( G e ( A n )) = 1 if and only if one of n, n − , n − , n/ , ( n − / , ( n − / is prime.Proof. By Lemma 4.7, we have G ∗ e ( S n ) = G ∗∗ e ( S n ) . Now, we are done using Theorem 2.8and Theorem 4.6. (cid:3)
Acknowledgement.
The First author would like to thank Prof. Arvind Ayyer for hisconstant support and encouragement. The second author would like to thank Prof. Sivara-makrishnan Sivasubramanian for his constant support and encouragement. The third au-thor would like to thank Prof. Basudeb Datta for his constant support and encourage-ment. The first author was supported by Department of Science and Technology grantEMR/2016/006624 and partly supported by UGC Centre for Advanced Studies. Also thefirst author was supported by NBHM Post Doctoral Fellowship grant 0204/52/2019/RD-II/339. The second author was supported by a CSIR-SPM fellowship. The third author wassupported by NBHM Post Doctoral Fellowship grant 0204/3/2020/RD-II/2470.
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E-mail address : [email protected] (Hiranya Kishore Dey) Department of Mathematics, Indian Institute of Technology, Bom-bay, India.
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