The intersection graph of a group

2015 ◽  
Vol 14 (05) ◽  
pp. 1550065 ◽  
Author(s):  
S. Akbari ◽  
F. Heydari ◽  
M. Maghasedi
Keyword(s):  
Disjoint Union ◽  
Abelian Groups ◽  
Solvable Groups ◽  
Intersection Graph ◽  
Free Graph ◽  
Intersection Graphs ◽  
Cyclic Groups ◽  
Vertex Set

Let G be a group. The intersection graph of G, denoted by Γ(G), is the graph whose vertex set is the set of all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent if and only if H ∩ K ≠ 1. In this paper, we show that the girth of Γ(G) is contained in the set {3, ∞}. We characterize all solvable groups whose intersection graphs are triangle-free. Moreover, we show that if G is finite and Γ(G) is triangle-free, then G is solvable. Also, we prove that if Γ(G) is a triangle-free graph, then it is a disjoint union of some stars. Among other results, we classify all abelian groups whose intersection graphs are complete. Finally, we study the intersection graphs of cyclic groups.

scholarly journals Representations of Edge Intersection Graphs of Paths in a Tree

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Martin Charles Golumbic ◽  
Marina Lipshteyn ◽  
Michal Stern
Keyword(s):  
Analogous Result ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Network Applications ◽  
Host Trees ◽  
Complete Problems ◽  
Vertex Set ◽  
International Audience ◽  
Np Complete ◽  
Simple Paths

International audience Let $\mathcal{P}$ be a collection of nontrivial simple paths in a tree $T$. The edge intersection graph of $\mathcal{P}$, denoted by EPT($\mathcal{P}$), has vertex set that corresponds to the members of $\mathcal{P}$, and two vertices are joined by an edge if the corresponding members of $\mathcal{P}$ share a common edge in $T$. An undirected graph $G$ is called an edge intersection graph of paths in a tree, if $G = EPT(\mathcal{P})$ for some $\mathcal{P}$ and $T$. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs.

Intersection graphs of general linear groups

2020 ◽  
pp. 2150039
Author(s):  
Mai Hoang Bien ◽  
Do Hoang Viet
Keyword(s):  
Finite Field ◽  
Linear Group ◽  
General Linear Group ◽  
Undirected Graph ◽  
Intersection Graph ◽  
General Linear ◽  
Linear Groups ◽  
Intersection Graphs ◽  
General Linear Groups ◽  
Vertex Set

Let [Formula: see text] be a field and [Formula: see text] the general linear group of degree [Formula: see text] over [Formula: see text]. The intersection graph [Formula: see text] of [Formula: see text] is a simple undirected graph whose vertex set includes all nontrivial proper subgroups of [Formula: see text]. Two vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] is a finite field containing at least three elements, then the diameter [Formula: see text] is [Formula: see text] or [Formula: see text]. We also classify [Formula: see text] according to [Formula: see text]. In case [Formula: see text] is infinite, we prove that [Formula: see text] is one-ended of diameter 2 and its unique end is thick.

SOME RESULTS ON THE INTERSECTION GRAPHS OF IDEALS OF RINGS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250200 ◽  
Author(s):  
S. AKBARI ◽  
R. NIKANDISH ◽  
M. J. NIKMEHR
Keyword(s):  
Chromatic Number ◽  
Clique Number ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Graph Theoretic ◽  
Reduced Ring ◽  
Algebraic Properties ◽  
Vertex Set

Let R be a ring with unity and I(R)* be the set of all nontrivial left ideals of R. The intersection graph of ideals of R, denoted by G(R), is a graph with the vertex set I(R)* and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ 0. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose intersection graphs of ideals are not connected. Also we determine all rings whose clique number of the intersection graphs of ideals is finite. Among other results, it is shown that for a ring R, if the clique number of G(R) is finite, then the chromatic number is finite and if R is a reduced ring, then both are equal.

scholarly journals Structure of intersection graphs

2021 ◽  
Vol 5 (2) ◽  
pp. 102
Author(s):  
Haval M. Mohammed Salih ◽  
Sanaa M. S. Omer
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Special Linear Group ◽  
Intersection Graph ◽  
Graph Structure ◽  
Normal Subgroups ◽  
Intersection Graphs ◽  
Cyclic Groups ◽  
A Finite Group

<p style="text-align: left;" dir="ltr"> Let <em>G</em> be a finite group and let <em>N</em> be a fixed normal subgroup of <em>G</em>.  In this paper, a new kind of graph on <em>G</em>, namely the intersection graph is defined and studied. We use <img src="/public/site/images/ikhsan/equation.png" alt="" width="6" height="4" /> to denote this graph, with its vertices are all normal subgroups of <em>G</em> and two distinct vertices are adjacent if their intersection in <em>N</em>. We show some properties of this graph. For instance, the intersection graph is a simple connected with diameter at most two. Furthermore we give the graph structure of <img src="/public/site/images/ikhsan/equation_(1).png" alt="" width="6" height="4" /> for some finite groups such as the symmetric, dihedral, special linear group, quaternion and cyclic groups. </p>

Planarity of the intersection graph of subgroups of a finite group

2016 ◽  
Vol 15 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Hadi Ahmadi ◽  
Bijan Taeri
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Finite Groups ◽  
Prime Order ◽  
Undirected Graph ◽  
Solvable Groups ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
A Finite Group

For a nontrivial finite group [Formula: see text] different from a cyclic group of prime order, the intersection graph [Formula: see text] of [Formula: see text] is the simple undirected graph whose vertices are the nontrivial proper subgroups of [Formula: see text] and two vertices are joined by an edge if and only if they have a nontrivial intersection. In this paper we characterize all finite groups with planar intersection graphs. It turns out that few solvable groups have planar intersection graphs. Also we classify finite groups whose intersection graphs are bipartite, triangle free and forests.

scholarly journals Decomposable Twofold Triple Systems with Non-Hamiltonian 2-Block Intersection Graphs

10.37236/7969 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Rosalind A. Cameron ◽  
David A. Pike
Keyword(s):  
Triple System ◽  
Hamilton Cycle ◽  
Intersection Graph ◽  
Steiner Triple Systems ◽  
Intersection Graphs ◽  
Triple Systems ◽  
Counter Example ◽  
Vertex Set ◽  
Block Intersection ◽  
Twofold Triple System

The $2$-block intersection graph ($2$-BIG) of a twofold triple system (TTS)  is the graph whose vertex set is composed of the blocks of the TTS and two vertices are joined by an edge if the corresponding blocks intersect in exactly two elements. The $2$-BIGs are themselves interesting graphs: each component is cubic and $3$-connected, and a $2$-BIG is bipartite exactly when the TTS is decomposable to two Steiner triple systems. Any connected bipartite $2$-BIG with no Hamilton cycle is a further counter-example to a disproved conjecture posed by Tutte in 1971. Our main result is that there exists an integer $N$ such that for all $v\geq N$, if  $v\equiv 1$ or $3\mod{6}$ then there exists a TTS($v$) whose $2$-BIG is bipartite and connected but not Hamiltonian. Furthermore, $13<N\leq 663$. Our approach is to construct a TTS($u$) whose $2$-BIG is connected bipartite and non-Hamiltonian and embed it within a TTS($v$)  where $v>2u$ in such a way that, after a single trade, the $2$-BIG of the resulting TTS($v$) is bipartite connected and non-Hamiltonian.

scholarly journals The Largest Component in an Inhomogeneous Random Intersection Graph with Clustering

10.37236/382 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Mindaugas Bloznelis
Keyword(s):  
Probability Measure ◽  
Branching Process ◽  
Intersection Graph ◽  
Connected Component ◽  
Intersection Graphs ◽  
First Order ◽  
Random Intersection Graph ◽  
Vertex Set

Given integers $n$ and $m=\lfloor\beta n \rfloor$ and a probability measure $Q$ on $\{0, 1,\dots, m\}$, consider the random intersection graph on the vertex set $[n]=\{1,2,\dots, n\}$ where $i,j\in [n]$ are declared adjacent whenever $S(i)\cap S(j)\neq\emptyset$. Here $S(1),\dots, S(n)$ denote the iid random subsets of $[m]$ with the distribution $\bf{P}(S(i)=A)={{m}\choose{|A|}}^{-1}Q(|A|)$, $A\subset [m]$. For sparse random intersection graphs, we establish a first-order asymptotic as $n\to \infty$ for the order of the largest connected component $N_1=n(1-Q(0))\rho+o_P(n)$. Here $\rho$ is the average of nonextinction probabilities of a related multitype Poisson branching process.

scholarly journals Remark on subgroup intersection graph of finite abelian groups

2020 ◽  
Vol 18 (1) ◽  
pp. 1025-1029
Author(s):  
Jinxing Zhao ◽  
Guixin Deng
Keyword(s):  
Finite Group ◽  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Intersection Graph ◽  
Finite Abelian Groups ◽  
Intersection Graphs ◽  
A Finite Group ◽  
Identity Elements

Abstract Let G be a finite group. The subgroup intersection graph \text{&#x0393;}(G) of G is a graph whose vertices are non-identity elements of G and two distinct vertices x and y are adjacent if and only if |\langle x\rangle \cap \langle y\rangle |\gt 1 , where \langle x\rangle is the cyclic subgroup of G generated by x. In this paper, we show that two finite abelian groups are isomorphic if and only if their subgroup intersection graphs are isomorphic.

scholarly journals Turán-type results for intersection graphs of boxes

2021 ◽  
pp. 1-6
Author(s):  
István Tomon ◽  
Dmitriy Zakharov
Keyword(s):  
Short Note ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Poset Dimension ◽  
Axis Parallel ◽  
Turán Theorem ◽  
Separation Dimension

Abstract In this short note, we prove the following analog of the Kővári–Sós–Turán theorem for intersection graphs of boxes. If G is the intersection graph of n axis-parallel boxes in $${{\mathbb{R}}^d}$$ such that G contains no copy of K t,t , then G has at most ctn( log n)2d+3 edges, where c = c(d)>0 only depends on d. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension. Using this approach, we also show that a construction of Basit, Chernikov, Starchenko, Tao and Tran of K2,2-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon, Basavaraju, Chandran, Mathew and Rajendraprasad. We show that there exist graphs of separation dimension 4 having superlinear number of edges.

scholarly journals On connectedness of power graphs of finite groups

2018 ◽  
Vol 17 (10) ◽  
pp. 1850184 ◽  
Author(s):  
Ramesh Prasad Panda ◽  
K. V. Krishna
Keyword(s):  
Finite Groups ◽  
Upper Bound ◽  
The Other ◽  
Number Of Components ◽  
Cyclic Groups ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Power

The power graph of a group [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of [Formula: see text]-groups are studied. In particular, the number of components of that of abelian [Formula: see text]-groups are determined.

Sign in / Sign up

Export Citation Format

Share Document