The purpose of this note is to define a graph whose vertex set is a finite group $G$, whose edge set is contained in that of the commuting graph of $G$ and contains the enhanced power graph of $G$. We call this graph the deep commuting graph of $G$. Two elements of $G$ are joined in the deep commuting graph if and only if their inverse images in every central extension of $G$ commute.
We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that the automorphism group of $G$ acts as automorphisms of the deep commuting graph.
We prove a formal power series identity, relating the arithmetic sum-of-divisors function to commuting triples of permutations. This establishes a conjecture of Franklin T. Adams-Watters.
Let $G$ be a non-abelian group and $Z(G)$ be its center. The non-commuting graph $\mathcal{A}_G$ of $G$ is the graph whose vertex set is $G\backslash Z(G)$ and two vertices are joined by an edge if they do not commute. Let $\mathrm{SL}(2,q)$ be the special linear group of degree 2 over the finite field of order $q$. In this paper we prove that if $G$ is a group such that $\mathcal{A}_G\cong \mathcal{A}_{\mathrm{SL}(2,q)}$ for some prime power $q\geq 2$, then $G\cong \mathrm{SL}(2,q)$.
The commuting graph of a finite non-commutative semigroup $S$, denoted $\cg(S)$, is a simple graph whose vertices are the non-central elements of $S$ and two distinct vertices $x,y$ are adjacent if $xy=yx$. Let $\mi(X)$ be the symmetric inverse semigroup of partial injective transformations on a finite set $X$. The semigroup $\mi(X)$ has the symmetric group $\sym(X)$ of permutations on $X$ as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of $\sym(X)$. In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of $\cg(\sym(X))$…
Let $G$ be a group. The \emph{power graph} of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chromatic number. Furthermore, it is proved that for every group $G$, the clique number of the power graph of $G$ is at most countably infinite. We also measure how close the power graph is to the \emph{commuting graph} by introducing a new graph which …
In this paper, we consider various graphs, namely: power graph, cyclic graph, enhanced power graph and commuting graph, on a finite semigroup $S$. For an arbitrary pair of these four graphs, we classify finite semigroups such that the graphs in this pair are equal. In this connection, for each of the graph we also give a necessary and sufficient condition on $S$ such that it is complete. The work of this paper generalize the corresponding results obtained for groups.
The enhanced power graph of a group $G$ is a simple undirected graph with vertex set $G$ and two vertices are adjacent if they belong to same cyclic subgroup. In this paper, we study distant properties and detour distant properties such as closure, interior, distance degree sequence and eccentric subgraph of the enhanced power graph of semidihedral group. Consequently, we obtained the metric dimension and resolving polynomial of the enhanced power graph of semidihedral group. At the final part of this paper, we obtained the Laplacian spectrum of the enhanced power graph of semidihedral, dihedr…
A graph is weakly perfect if its clique number and chromatic number are equal. We show that the enhanced power graph of a finite group $G$ is weakly perfect: its clique number and chromatic number are equal to the maximum order of an element of $G$. The proof requires a combinatorial lemma. We give some remarks about related graphs.
The enhanced power graph of a finite group $G$, denoted by $\mathcal{P}_E(G)$, is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. In this article, we determine all finite groups such that the minimum degree and the vertex connectivity of $\mathcal{P}_E(G)$ are equal. Also, we classify all groups whose (proper) enhanced power graphs are strongly regular. Further, the vertex connectivity of the enhanced power graphs associated to some nilpotent groups is obtained. Finally, we obtain a lower boun…
Given a group $G$, the enhanced power graph of $G$ denoted by $\mathcal{G}_e(G)$, is the graph with vertex set $G$ and two distinct vertices $x, y$ are edge connected in $\mathcal{G}_e(G)$ if there exists $z\in G $ such that $x=z^m$ and $ y=z^n $, for some $m, n\in \mathbb{N}$. In this article, we characterize the enhanced power graph $\mathcal{G}_e(G)$ of $G$. The graph $\mathcal{G}_e(G)$ is complete if and only if $G$ is cyclic, and $\mathcal{G}_e(G)$ is Eulerian if and only if $|G|$ is odd. We classify all abelian groups and also all non-abelian $p-$groups $G$ for which $\mathcal{G}_e(G)$ s…
The enhanced power graph $\mathcal{P}_e(G)$ of a group $G$ is a graph with vertex set $G$ and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, independence number and matching number of enhanced power graphs of finite groups. We first study these graph invariants for $\mathcal{P}_e(G)$ when $G$ is any finite group, and then determine them when $G$ is a finite abelian $p$-group, $U_{6n} = \langle a, b : a^{2n} = b^3 = e, ba =ab^{-1} \rangle$, the dihedral group $D_{2n}$, or the semidihedral group $SD_{8n}$. If $G$ is any of the…
The enhanced power graph of a finite group $G$ is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. An $L( 2,1)$-labeling of graph $Γ$ is an integer labeling of $V(Γ)$ such that adjacent vertices have labels that differ by at least $2$ and vertices distance $2$ apart have labels that differ by at least $1$. The $λ$-number of $Γ$, denoted by $λ(Γ)$, is the minimum range over all $L( 2,1)$-labelings. In this article, we study the lambda number of the enhanced power graph $\mathcal{P}_E(G)$ of the …
For any positive integer $m$, the complete graph on $2^{2m}(2^m+2)$ vertices is decomposed into $2^m+1$ commuting strongly regular graphs, which give rise to a symmetric association scheme of class $2^{m+2}-2$. Furthermore, the eigenmatrices of the symmetric association schemes are determined explicitly. As an application, the eigenmatrix of the commutative strongly regular decomposition obtained from the strongly regular graphs is derived.
For a group $G,$ the enhanced power graph of $G$ is a graph with vertex set $G$ in which two distinct elements $x, y$ are adjacent if and only if there exists an element $w$ in $G$ such that both $x$ and $y$ are powers of $w.$ The proper enhanced power graph is the induced subgraph of the enhanced power graph on the set $G \setminus S,$ where $S$ is the set of dominating vertices of the enhanced power graph. In this paper, we first characterize the dominating vertices of enhanced power graph of any finite nilpotent group. Thereafter, we classify all nilpotent groups $G$ such that the proper en…
This paper deals with the vertex connectivity of enhanced power graph of finite group. We classify all abelian groups G such that vertex connectivity of enhanced power graph of G is 1. We derive an upper bound of vertex connectivity for the enhanced power graph 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 special class of non-abelian group G such that the proper enhanced power graph is connected and we find their vertex connectivity.
The enhanced power graph $\mathcal G_e(\mathbf G)$ of a group $\mathbf G$ is the graph with vertex set $G$ such that two vertices $x$ and $y$ are adjacent if they are contained in a same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between power graphs of finite groups is an isomorhism between enhanced power graphs of these groups, and we find all finite groups $\mathbf G$ for which $\mathrm{Aut}(\mathcal G_e(\mathbf G)$ is abelian, all finite groups $\mathbf G$ with $\lvert\mathrm{Aut}(\m…
Let $G$ be a finite group, the enhanced power graph of $G$, denoted by $Γ_G^e$, is the graph with vertex set $G$ and two vertices $x,y$ are edge connected in $Γ_{G}^e$ if there exist $z\in G$ such that $x,y\in\langle z\rangle$. Let $ζ$ be a edge-coloring of $Γ_G^e$. In this article, we calculate the rainbow connection number of the enhanced power graph $Γ_G^e$.
The directed power graph $\vec{\mathcal P}(\mathbf G)$ of a group $\mathbf G$ is the simple digraph with vertex set $G$ such that $x\rightarrow y$ if $y$ is a power of $x$. The power graph of $\mathbf G$, denoted by $\mathcal P(\mathbf G)$, is the underlying simple graph. The enhanced power graph $\mathcal P_e(\mathbf G)$ of $\mathbf G$ is the simple graph with vertex set $G$ in which two elements are adjacent if they generate a cyclic subgroup. In this paper, it is proven that, if two groups have isomorphic power graphs, then they have isomorphic enhanced power graphs, too. It is known that a…
