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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…

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…

Consider a plane graph G, drawn with straight lines. For every pair a,b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual distance in the plane. The worst-case ratio…

The dominating graph of a graph G is a graph whose vertices correspond to the dominating sets of G and two vertices are adjacent whenever their corresponding dominating sets differ in exactly one vertex. Studying properties of dominating…

We give a sharp bound on the number of triangles in a graph with fixed number of edges. We also characterize graphs that achieve the maximum number of triangles. Using the upper bound on number of triangles, we prove that if $G$ is a…

Let $G$ be a finite, simple connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The remoteness $\rho(G)$ of $G$ is the maximum of the average distances…

In this article we introduce the zero-divisor graphs $\Gamma_\mathscr{P}(X)$ and $\Gamma^\mathscr{P}_\infty(X)$ of the two rings $C_\mathscr{P}(X)$ and $C^\mathscr{P}_\infty(X)$; here $\mathscr{P}$ is an ideal of closed sets in $X$ and…

Let $k$ be an integer. We prove a rough structure theorem for separations of order at most $k$ in finite and infinite vertex transitive graphs. Let $G = (V,E)$ be a vertex transitive graph, let $A \subseteq V$ be a finite vertex-set with…

Monadically stable and monadically NIP classes of structures were initially studied in the context of model theory and defined in logical terms. They have recently attracted attention in the area of structural graph theory, as they…

Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $\lambda (G)$ as the minimum number of nonsoluble factors in a series of…

Given a finite group $G$, the generating graph $\Gamma(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we…

The proper commuting graph $\mathcal{C}^{**}(G)$ of a finite group $G$ is the simple graph whose vertices are the noncentral elements of $G$ and two distinct vertices are adjacent if they commute. In this paper, we study the domination…

Let $\Gamma$ be a locally finite graph, $L$ the normalized Laplacian of $\Gamma$. If $\Gamma$ is uniformy locally finite, i.e. if each vertex has no more than $d$ adjacent vertices, then the matrix of $L$ (with respect to the standard…

A partition P of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of P. The partition dimension of G is the minimum cardinality of a…

The conditional diameter of a connected graph $\Gamma=(V,E)$ is defined as follows: given a property ${\cal P}$ of a pair $(\Gamma_1, \Gamma_2)$ of subgraphs of $\Gamma$, the so-called \emph{conditional diameter} or ${\cal P}$-{\em…

A set of vertices $S$ \emph{resolves} a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \emph{metric dimension} of $G$ is the minimum cardinality of a resolving set of $G$.…

The unit ball random geometric graph $G=G^d_p(\lambda,n)$ has as its vertices $n$ points distributed independently and uniformly in the $d$-dimensional unit ball, with two vertices adjacent if and only if their $l_p$-distance is at most…

For a graph $\Gamma$ and group $G$, $G^\Gamma$ is the subgroup of $G^{|\Gamma|}$ generated by elements with $g$ in the coordinates corresponding to $v$ and its neighbors in $\Gamma$. There is a natural epimorphism $G^\Gamma \to…

Let $H$ be a normal subgroup of a group $G$. The normal subgroup based power graph $\Gamma_H(G)$ of $G$ is the simple undirected graph with vertex set $V(\Gamma_H(G))= (G\setminus H)\cup \{e\}$ and two distinct vertices $a$ and $b$ are…

The general position problem is to find the cardinality of a largest vertex subset S such that no triple of vertices of S lie on a common geodesic. For a connected graph G, the cardinality of S is denoted by gp(G) and called gp-number (or…