Related papers: A New Characterization of Sporadic Groups

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…

The prime graph of a finite group $G$, which is denoted by ${\rm GK}(G)$, is a simple graph whose vertex set is comprised of the prime divisors of $|G|$ and two distinct prime divisors $p$ and $q$ are joined by an edge if and only if there…

Let G be a finite group with identity e and H \neq \{e\} be a subgroup of G. The generalized non-coprime graph GAmma_{G,H} of G with respect to H is the simple undirected graph with G - \{e \}\) as the vertex set and two distinct vertices a…

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…

The {\em distinguishing number} of a group $G$ acting faithfully on a set $V$ is the least number of colors needed to color the elements of $V$ so that no non-identity element of the group preserves the coloring. The {\em distinguishing…

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple graph with vertex set $G$, in which two distinct vertices are adjacent if one of them is a power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group…

Let $G$ be a finite group. Denoting by ${\rm{cd}}(G)$ the set of the degrees of the irreducible complex characters of $G$, we consider the {\it character degree graph} of $G$: this is the (simple, undirected) graph whose vertices are the…

We introduce a directed graph related to a group $G$, which we call the N-prime graph $\Gamma_{\rm{N}}(G)$ of $G$ and which is a refinement of the classical Gruenberg-Kegel graph. The vertices of $\Gamma_{\rm{N}}(G)$ are the primes $p$ such…

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a…

Let $G$ be a group. The intersection subgroup graph of $G$ (introduced by Anderson et al. \cite{anderson}) is the simple graph $\Gamma_{S}(G)$ whose vertices are those non-trivial subgroups say $H$ of $G$ with $H\cap K=\{e\}$ for some…

The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring $\mathbb Z G$ , i.e. the prime graph of the normalised unit group of $\mathbb Z G$ coincides with that one of the group $G$. In this note we prove for…

For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes the path-length distance. Describing the set $\pi(G)$ consisting of those $q\in \mathbb{R}$…

Two finite groups are said to be isospectral if they have the same sets of element orders. This paper completes the description of finite groups that are isospectral to finite simple groups with disconnected prime graph.

The cyclic graph $\Gamma(S)$ of a semigroup $S$ is the simple graph whose vertex set is $S$ and two vertices $x, y$ are adjacent if the subsemigroup generated by $x$ and $y$ is monogenic. In this paper, we classify the semigroup $S$ such…

The power graph $\mathcal{P}(G)$ is a graph with group elements as vertex set and two elements are adjacent if one is a power of the other. The order supergraph $\mathcal{S}(G)$ of the power graph $\mathcal{P}(G)$ is a graph with vertex set…

Let $G$ be a finite group. The co-prime order graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices $x,y$ are adjacent if gcd$(o(x),o(y))$ is either $1$ or a prime, where $o(x)$ and $o(y)$ are the orders of $x$ and…

The commuting graph of a group $G$ is the graph whose vertices are the elements of $G$, two distinct vertices joined if they commute. Our purpose in this paper is twofold: we discuss the computational problem of deciding whether a given…

For a finite group $G$ the co-prime graph $\Gamma(G)$ is defined as a graph with vertex set $G$ in which two distinct vertices $x$ and $y$ are adjacent if and only if $gcd(o(x),o(y))=1$ where $o(x)$ and $o(y)$ denote the orders of the…

The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of elements of $G$. We present a polynomial-time algorithm that, given a finite set $\mathcal M$ of positive integers, outputs either an empty set or a finite simple group…

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…