On the power graph of a finite group

The power graph $\mathcal P_G$ of a finite group $G$ is the graph with the vertex set $G$, where two elements are adjacent if one is a power of the other. We first show that $\mathcal P_G$ has an transitive orientation, so it is a perfect graph and its core is a complete graph. Then we use the poset on all cyclic subgroups (under usual inclusion) to characterise the structure of $\mathcal P_G$. Finally, the closed formula for the metric dimension of $\mathcal P_G$ is established. As an application, we compute the metric dimension of the power graph of a cyclic group

The number of spanning trees of power graphs associated with specific groups and some applications

Given a group $G$, we define the power graph $\mathcal{P}(G)$ as follows: the vertices are the elements of $G$ and two vertices $x$ and $y$ are joined by an edge if $\langle x\rangle\subseteq \langle y\rangle$ or $\langle y\rangle\subseteq \langle x\rangle$. Obviously the power graph of any group is always connected, because the identity element of the group is adjacent to all other vertices. In the present paper, among other results, we will find the number of spanning trees of the power graph associated with specific finite groups. We also determine, up to isomorphism, the structure of a finite group $G$ whose power graph has exactly $n$ spanning trees, for $n<5^3$. Finally, we show that the alternating group $\mathbb{A}_5$ is uniquely determined by tree-number of its power graph among all finite simple groups.Comment: 28 pages, Ars Combinatoria, 201

Power graphs of (non)orientable genus two

The power graph $\Gamma_G$ of a finite group $G$ is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this paper, we classify the finite groups whose power graphs have (non)orientable genus two.Comment: 17 pages, 7 figure

A description of automorphism group of power graphs of finite groups

The power graph of a group is the graph whose vertex set is the set of nontrivial elements of group, two elements being adjacent if one is a power of the other. We introduce some way for find the automorphism groups of some graphs. As an application We describe the full automorphism group of the power graph of all finite groups. Also we obtain the full automorphism group of power graph of abelian, homocyclic and nilpotent groupsComment: 7 page

Lambda number of the power graph of a finite group

The power graph $\Gamma_G$ of a finite group $G$ is the graph with the vertex set $G$, where two distinct elements are adjacent if one is a power of the other. An $L(2, 1)$-labeling of a graph $\Gamma$ is an assignment of labels from nonnegative integers to all vertices of $\Gamma$ such that vertices at distance two get different labels and adjacent vertices get labels that are at least $2$ apart. The lambda number of $\Gamma$, denoted by $\lambda(\Gamma)$, is the minimum span over all $L(2, 1)$-labelings of $\Gamma$. In this paper, we obtain bounds for $\lambda(\Gamma_G)$, and give necessary and sufficient conditions when the bounds are attained. As applications, we compute the exact value of $\lambda(\Gamma_G)$ if $G$ is a dihedral group, a generalized quaternion group, a $\mathcal{P}$-group or a cyclic group of order $pq^n$, where $p$ and $q$ are distinct primes and $n$ is a positive integer.Comment: 13 pages, 1 figur

Certain properties of the power graph associated with a finite group

There are a variety of ways to associate directed or undirected graphs to a group. It may be interesting to investigate the relations between the structure of these graphs and characterizing certain properties of the group in terms of some properties of the associated graph. The power graph $\mathcal{P}(G)$ of a group $G$ is a simple graph whose vertex-set is $G$ and two vertices $x$ and $y$ in $G$ are adjacent if and only if $y=x^m$ or $x=y^m$ for some positive integer $m$. We also pay attention to the subgraph $\mathcal{P}^\ast(G)$ of $\mathcal{P}(G)$ which is obtained by deleting the vertex 1 (the identity element of $G$). In the present paper, we first investigate some properties of the power graph $\mathcal{P}(G)$ and the subgraph $\mathcal{P}^\ast(G)$. We next prove that many of finite groups such as finite simple groups, symmetric groups and the automorphism groups of sporadic simple groups can be uniquely determined by their power graphs among all finite groups. We have also determined up to isomorphism the structure of any finite group $G$ such that the graph $\mathcal{P}^\ast(G)$ is a strongly regular graph, a bipartite graph, a planar graph or an Eulerian graph. Finally, we obtained some infinite families of finite groups such that the graph $\mathcal{P}^\ast(G)$ containing some cut-edges.Comment: 22 page

Enhanced Power Graphs of Finite Groups

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 $\lvert\mathrm{Aut}(\mathcal G_e(\mathbf G)\rvert$ being prime power, and all finite groups $\mathbf G$ with $\lvert\mathrm{Aut}(\mathcal G_e(\mathbf G)\rvert$ being square free. Also we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph

On some characterizations of strong power graphs of finite groups

Let $G$ be a finite group of order $n$. The strong power graph $\mathcal{P}_s(G)$ of $G$ is the undirected graph whose vertices are the elements of $G$ such that two distinct vertices $a$ and $b$ are adjacent if $a^{{m}_1}$=$b^{{m}_2}$ for some positive integers ${m}_1 ,{m}_2 < n$. In this article we classify all groups $G$ for which $\mathcal{P}_s(G)$ is line graph and Caley graph. Spectrum and permanent of the Laplacian matrix of the strong power graph $\mathcal{P}_s(G)$ are found for any finite group $G$.Comment: 13 page

A combinatorial characterization of finite groups of prime exponent

The power graph of a group $G$ is a simple and undirected graph with vertex set $G$ and two distinct vertices are adjacent if one is a power of the other. In this article, we characterize (non-cyclic) finite groups of prime exponent and finite elementary abelian $2$-groups (of rank at least $2$) in terms of their power graphs

The diameter of proper power graphs of alternating groups

The power graph of finite group G is a simple graph whose vertex set is G and two distinct elements a and b are adjacent if and only if one of them is a power of the other. The proper power graph of G is a graph which is obtained by deleting the identity vertex (the identity element of G). In this paper, we improve the diameter bound of proper power graph of alternating group of degree n which the graph is connected. We show that the diameter of An is between 6 and 11, if the n at least 51. We also describe a number of short paths in these power graphs.Comment: 9 page