相关论文: Generalizing the enhanced power graph of a group w…

These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…

The power graph of a group $G$ is a graph with vertex set $G$, in which two vertices are adjacent if one is some power of the other. In the commuting graph, with $G$ as the vertex set, two vertices are joined by an edge if they commute in…

The enhanced power graph of a group $G$ is the graph $P_e(G)$ whose vertex set is $G$, such that two distinct vertices $x$ and $y$, are adjacent if $\langle x, y\rangle$ is cyclic. In this paper, we analyze the structure of the enhanced…

Let $G$ be a group. The directed endomorphism graph, \dend of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex `$a$' to the vertex `$\, b$' $(a \neq b) $ if and only if there exists an endomorphism on…

Let $G$ be a finite group. Define a graph on the set $G^{\#} = G \setminus \{ 1 \}$ by declaring distinct elements $x,y\in G^{\#}$ to be adjacent if and only if $\langle x,y\rangle$ is cyclic. Denote this graph by $\Delta(G)$. The graph…

We determine the diameter of every connected component of the complement of the power graph and the enhanced power graph of a finite group, which completely answers two questions by Peter J. Cameron.

Let $G$ be a group. The directed endomorphism graph, $\dend(G)$ of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex $a$ to the vertex $b$ if $a \neq b$ and there exists an endomorphism on $G$ mapping…

The (proper) power graph of a group is a graph whose vertex set is the set of all (nontrivial) elements of the group and two distinct vertices are adjacent if one is a power of the other. Various kinds of planarity of (proper) power 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…

Let $G$ be a group. The power graph of $G$ is a graph with vertex set $G$ in which two distinct elements $x,y$ are adjacent if one of them is a power of the other. We characterize all groups whose power graphs have finite independence…

The power graph of a group is the graph whose vertex set is the set of non-trivial elements of group, two elements being adjacent if one is a power of the other. We define a new power graph and study on connectivity, diameter and clique…

Let $G$ be a finite group. A number of graphs with the vertex set $G$ have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful…

The power graph of a group is the simple graph whose vertices are the group elements and two vertices are adjacent whenever one of them is a positive power of the other. We characterize the finite nilpotent groups whose power graphs have…

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…

Let $G$ be a finite group and construct a graph $\Delta(G)$ by taking $G\setminus\{1\}$ as the vertex set of $\Delta(G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$ be the set…

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…

The two graphs of the title both have vertex set G. In the intersection power graph, x and y are joined if some non-identity element is a power of both; in the power graph, x and y joined if one is a power of the other. Thus the power graph…

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…

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…

Assume that there is a free group action of automorphisms on a bipartite graph. If there is a perfect matching on the factor graph, then obviously there is a perfect matching on the graph. Surprisingly, the reversed is also true for…