In this paper, for a finite group, we investigate to what extent its directed (resp. undirected) reduced power graph determines its directed power graph (resp. reduced power graph). Moreover, we in...

On the power graph and the reduced power graph of a finite group

The reduced power graph ℛ&#x1d4ab;(G) of a group G is the graph whose vertex set is the set of all elements of G, and two vertices u and v are adjacent in ℛ&#x1d4ab;(G) if and only if 〈u〉⊂〈v〉 or 〈v〉⊂〈u〉. The prope...

Characterization of groups with planar, toroidal or projective planar (proper) reduced power graphs

Let $G$ be a finite group. The order supergraph of $G$ is the graph with vertex set $G$, and two distinct vertices $x,y$ are adjacent if $o(x)\mid o(y)$ or $o(y)\mid o(x)$. The enhanced power graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices are adjacent if they generate a cyclic subgroup. The reduced power graph of $G$ is the graph with vertex set $G$, and two distinct vertices $x,y$ are adjacent if $\langle x\rangle \subset \langle y\rangle$ or $\langle y\rangle \subset \langle x\rangle$. In this paper, we characterize the strong metric dimension of the order supergraph, the enhanced power graph and the reduced power graph of a finite group.

Strong metric dimensions for power graphs of finite groups

The directed power graph $\vec{\mathcal G}(\mathbf G)$ 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 with $\mathcal G(\mathbf G)$, is the underlying simple graph. 
In this paper, for groups $\mathbf G$ and $\mathbf H$, the following is proved. If $\mathbf G$ has no quasicyclic subgroup $\mathbf C_{p^\infty}$ which has trivial intersection with every cyclic subgroup $\mathbf K$ of $\mathbf G$ such that $\mathbf K\not\leq\mathbf C_{p^\infty}$, then $\mathcal G(\mathbf G)\cong \mathcal G(\mathbf H)$ implies $\vec{\mathcal G}(\mathbf G)\cong \vec{\mathcal G}(\mathbf H)$. Consequently, any two torsion-free groups having isomorphic power graphs have isomorphic directed power graphs.

The Power Graph of a Torsion-Free Group Determines the Directed Power Graph

Let Γ be a graph with vertex set V(Γ). A subset C of V(Γ) is a perfect code of Γ if C is an independent set such that every vertex in V(Γ)∖C is adjacent to exactly one vertex in C. A subset T of V(...

R. Rajkumar

Papers

Reduced power graph of a group

Intersection graphs of cyclic subgroups of groups