Related papers: A survey on conjugacy class graphs of groups

A natural question for groups $H$ is which data can be detected in its finite quotients. A subset $X \subset H$ is called separable if for all $h\in H \setminus X$, there exists an epimorphism $\varphi$ to a finite group $Q$ such that…

This paper deals with the classification of groups $G$ such that power graphs and proper power graphs of $G$ are line graphs. In fact, we classify all finite nilpotent groups whose power graphs are line graphs. Also, we categorize all…

The power graph of a finite group $G$ is a simple undirected graph with vertex set $G$ and two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group $G$ is a simple undirected graph whose vertex…

We introduce and investigate the solvable graph $\Gamma_\mathfrak{S}(L)$ of a finite-dimensional Lie algebra $L$ over a field $F$. The vertices are the elements outside the solvabilizer $\sol(L)$, and two vertices are adjacent whenever they…

Given a finite group G, the bipartite divisor graph for its conjugacy class sizes is the bipartite graph with bipartition consisting of the set of conjugacy class sizes of G-Z (where Z denotes the centre of G) and the set of prime numbers…

Given a graph $G$, the $k$-dominating graph of $G$, $D_k(G)$, is defined to be the graph whose vertices correspond to the dominating sets of $G$ that have cardinality at most $k$. Two vertices in $D_k(G)$ are adjacent if and only if the…

The power graph $\mathcal{P}(G)$ of a finite group $G$ is a graph whose vertex set is the group $G$ and distinct elements $x,y\in G$ are adjacent if one is a power of the other, that is, $x$ and $y$ are adjacent if $x\in\langle y\rangle$ or…

We associate a graph $\Gamma_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | \left<x,y\right> \text{is cyclic for all} y\in G\}$, and…

Let $G$ be a classical group defined over a finite field. We consider the following fundamental problems concerning conjugacy in $G$: 1. List a representative for each conjugacy class of $G$. 2. Given $x \in G$, describe the centralizer of…

The processes of constructing some graphs from others using binary operations of union with intersection (gluing) are studied. For graph classes closed with respect to gluing operations the elemental and operational bases are introduced.…

We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of…

Let $\mathcal{C}$ be a class of groups. A group $X$ is said to be residually a $\mathcal{C}$-group (conjugacy $\mathcal{C}$-separable) if, for any elements $x,y \in X$ that are not equal (not conjugate in $X$), there exists a homomorphism…

Let $G=(V,E)$ be a graph. A set $S\subseteq V(G)$ is a dominating set, if every vertex in $V(G)\backslash S$ is adjacent to at least one vertex in $S$. The $k$-dominating graph of $G$, $D_k (G)$, is defined to be the graph whose vertices…

Let $G$ be a finite non-solvable group with solvable radical $Sol(G)$. The solvable graph $\Gamma_s(G)$ of $G$ is a graph with vertex set $G\setminus Sol(G)$ and two distinct vertices $u$ and $v$ are adjacent if and only if $\langle u, v…

For a simple graph $G=(V,E)$ with eigenvalues of the adjacency matrix $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$, the energy of the graph is defined by $E(G)=\sum_{j=1}^{n}|\lambda_{j}|$. Myriads of papers have been published in…

We show that various invariants of a unipotent conjugacy class in a connected semisimple group can be recovered purely in terms of data involving the Weyl group.

We describe the geometry of conjugation within any split subgroup $H$ of the full isometry group $G$ of $n$-dimensional Euclidean space. We prove that for any $h \in H$, the conjugacy class $[h]_H$ of $h$ is described geometrically by the…

Let $G$ be a graph and $A$ be its adjacency matrix. A graph $G$ is invertible if its adjacency matrix $A$ is invertible and the inverse of $G$ is a weighted graph with adjacency matrix $A^{-1}$. A signed graph $(G,\sigma)$ is a weighted…

We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility.

Graphs with given k vertices generate an (acyclic) simplicial complex. We describe the homology of its quotient complex, formed by all connected graphs, and demonstrate its applications to the topology of braid groups, knot theory,…