Related papers: Order divisor graphs of finite groups

There has been a great deal of attention recently to graphs whose vertex set is a group, defined using the group structure. (The commuting graph, where two elements are joined if they commute, is the oldest and most famous example.) The…

In this note, we show that among finite nilpotent groups of a given order or finite groups of a given odd order, the cyclic group of that order has the minimum number of edges in its cyclic subgroup graph. We also conjecture that this holds…

An internal or friendly partition of a vertex set $V(G)$ of a graph $G$ is a partition to two nonempty sets $A\cup B$ such that every vertex has at least as many neighbours in its own class as in the other one. Motivated by Diwan's…

A set of vertices in a graph is agile if, however we partition the set into two parts, we can always find two vertex-disjoint connected subgraphs where one covers the first and the other the second part. We present a characterization for…

Let $G=\Gamma(S)$ be a semigroup graph, i.e., a zero-divisor graph of a semigroup $S$ with zero element 0. For any adjacent vertices $x, y$ in $G$, denote $C(x,y)={z\in V(G) | N(z)={x,y}}$. Assume that in $G$ there exist two adjacent…

A graph is an efficient open (resp.\ closed) domination graph if there exists a subset of vertices whose open (resp.\ closed) neighborhoods partition its vertex set. Graphs that are efficient open as well as efficient closed (shortly EOCD…

Given a formation $\mathfrak F$, we consider the graph whose vertices are the elements of $G$ and where two vertices $g,h\in G$ are adjacent if and only if $\langle g,h \rangle \notin\mathfrak F$. We are interested in the two following…

Characterisations of interval graphs, comparability graphs, co-comparability graphs, permutation graphs, and split graphs in terms of linear orderings of the vertex set are presented. As an application, it is proved that interval graphs,…

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…

A finite group $G$ is said to be rational if every character of $G$ is rational-valued. The Gruenberg-Kegel graph of a finite group $G$ is the undirected graph whose vertices are the primes dividing the order of $G$ and the edges join…

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. The degree graph $\Delta(G)$ of $G$ is defined as the simple undirected graph whose vertex set ${\rm{V}}(G)$ consists…

Getting inspired by the famous no-three-in-line problem and by the general position subset selection problem from discrete geometry, the same is introduced into graph theory as follows. A set $S$ of vertices in a graph $G$ is a general…

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…

We introduce and study the concept which we call the splitting of a graph and compare algebraic properties of the edge ideals of graphs and those of their splitting graphs.

An arithmetical structure on a graph is given by a labeling of the vertices which satisfies certain divisibility properties. In this note, we look at several families of graphs and attempt to give counts on the number of arithmetical…

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…

For a graph $G$, the central graph $C(G)$ is the graph constructed from $G$ by subdividing each edge of $G$ with one vertex and also by adding an edge to every pair of non-adjacent vertices in $G$. Also for a graph $G$, let $\gamma(G)$ and…

In this paper, we investigate certain graphs defined on groups, with a focus on infinite groups. The graphs discussed are the power graph, the enhanced power graph, and the commuting graph whose vertex set is a group $G$. The power graph is…

In 1968, Erd\"os defined the Shift Graph as the graph whose vertices are the $k$-element subsets of $[n]=\{0,1,2,...,n-1\}$ such that $A=\{a_1,...,a_k\}$ and $B=\{b_1,...,b_k\}$ are neighbours iff $a_1<b_1=a_2<b_2=a_3<... <b_{n-1}=a_n<b_n$.…

To each finite frame $\varphi$ in an inner product space $\mathcal{H}$ we associate a simple graph $G(\varphi)$, called {\it frame graph}, with the vectors of the frame as vertices and there is an edge between vertices $f$ and $g$ provided…