Related papers: On groups whose conjugacy class sizes are not divi…

Given a finite group $G$, denote by $\Gamma(G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of $G$, and set two vertices of $\Gamma(G)$ to be adjacent if and only if they are not coprime…

Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on the set of conjugacy class sizes of $G$: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, two…

Given a finite group $G$ with a normal subgroup $N$, the simple graph $\Gamma_\textit{G}( \textit{N} )$ is a graph whose vertices are of the form $|x^G|$, where $x\in{N\setminus{Z(G)}}$, and $x^G$ is the $G$-conjugacy class of $N$…

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes…

Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on its set of conjugacy class sizes: this is the (simple undirected) graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, and two…

Let $G$ be a finite group and $N$ a normal subgroup of $G$. We determine the structure of $N$ when the graph $\Gamma_G(N)$, which is the graph associated to the conjugacy classes of $G$ contained in $N$, has no triangles and when the graph…

The commuting graph ${\Gamma(G)}$ of a group $G$ is the simple undirected graph with group elements as a vertex set and two elements $x$ and $y$ are adjacent if and only if $xy=yx$ in $G$. By eliminating the identity element of $G$ and all…

Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if…

In this paper we introduce the graph $\Gamma_{sc}(G)$ associated with a group $G$, called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of $G$ and two distinct conjugacy…

Given a finite group $G,$ we denote by $\Delta(G)$ the graph whose vertices are the elements $G$ and where two vertices $x$ and $y$ are adjacent if there exists a minimal generating set of $G$ containing $x$ and $y.$ We prove that…

The divisibility graph $D(G)$ for a finite group $G$ is a graph with vertex set $cs~(G)\setminus\{1\}$ where $cs~(G)$ is the set of conjugacy class sizes of $G$. Two vertices $a$ and $b$ are adjacent whenever $a$ divides $b$ or $b$ divides…

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…

The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$…

The {\it prime graph} $\Gamma(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order…

Let $G$ be a finite group. For some fixed prime $p$, let $\Gamma_p(G)$ be the common divisor graph built on the set of sizes of $p$-regular conjugacy classes of $G$: this is the simple undirected graph whose vertices are the class sizes of…

Let $G$ be a (finite or infinite) group such that $G/Z(G)$ is not simple. The non-commuting, non-generating graph $\Xi(G)$ of $G$ has vertex set $G \setminus Z(G)$, with vertices $x$ and $y$ adjacent whenever $[x,y] \ne 1$ and $\langle x, y…

The Divisibility Graph of a finite group $G$ has vertex set the set of conjugacy class lengths of non-central elements in $G$ and two vertices are connected by an edge if one divides the other. We determine the connected components of the…

To any finite group $G$, we may associate a graph whose vertices are the elements of $G$ and where two distinct vertices $x$ and $y$ are adjacent if and only if the order of the subgroup $\langle x, y\rangle$ is divisible by at least 3…

Assume that $G$ is a finite group and let $a$ and $b$ be non-negative integers. We define an undirected graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and…

Let $G$ be a finite group. The solubility graph associated with the finite group $G$, denoted by $\Gamma_{\cal S}(G)$, is a simple graph whose vertices are the non-trivial elements of $G$, and there is an edge between two distinct elements…