Birecognition of prime graphs, and minimal prime graphs

2021 ◽  
Author(s):  
Houmem Belkhechine ◽  
Cherifa Ben Salha ◽  
Pierre Ille
Keyword(s):  
Prime Graph ◽  
Prime Graphs ◽  
Minimal Prime

Given a graph [Formula: see text], a subset [Formula: see text] of [Formula: see text] is a module of [Formula: see text] if for each [Formula: see text], [Formula: see text] is adjacent to all the elements of [Formula: see text] or to none of them. For instance, [Formula: see text], [Formula: see text] and [Formula: see text] ([Formula: see text]) are the trivial modules of [Formula: see text]. A graph [Formula: see text] is prime if [Formula: see text] and all its modules are trivial. Given a prime graph [Formula: see text], consider [Formula: see text] such that [Formula: see text] is prime. Given a graph [Formula: see text] such that [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] are [Formula: see text]-similar if for each [Formula: see text], [Formula: see text] and [Formula: see text] are both prime or not. The graph [Formula: see text] is said to be [Formula: see text]-birecognizable if every graph, [Formula: see text]-similar to [Formula: see text], is prime. We study the graphs [Formula: see text] that are not [Formula: see text]-birecognizable, where [Formula: see text] such that [Formula: see text] is prime, by using the following notion of a minimal prime graph. Given a prime graph [Formula: see text], consider [Formula: see text] such that [Formula: see text] is prime. Given [Formula: see text], [Formula: see text] is [Formula: see text]-minimal if for each [Formula: see text] such that [Formula: see text], [Formula: see text] is not prime.

CHARACTERIZATION BY PRIME GRAPH OF PGL(2, pk) WHERE p AND k > 1 ARE ODD

2010 ◽  
Vol 20 (07) ◽  
pp. 847-873 ◽  
Author(s):  
Z. AKHLAGHI ◽  
B. KHOSRAVI ◽  
M. KHATAMI
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Prime Number ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
Prime Graphs ◽  
A Finite Group ◽  
Nonabelian Composition Factor

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.

scholarly journals On odd prime labeling of graphs

2020 ◽  
Vol 3 (3) ◽  
pp. 33-40
Author(s):  
Maged Zakaria Youssef ◽  
◽  
Zainab Saad Almoreed ◽  
Keyword(s):  
Necessary Conditions ◽  
Prime Graph ◽  
Prime Graphs ◽  
Vertex Set ◽  
Prime Labeling

In this paper we give a new variation of the prime labeling. We call a graph \(G\) with vertex set \(V(G)\) has an odd prime labeling if its vertices can be labeled distinctly from the set \(\big\{1, 3, 5, ...,2\big|V(G)\big| -1\big\}\) such that for every edge \(xy\) of \(E(G)\) the labels assigned to the vertices of \(x\) and \(y\) are relatively prime. A graph that admits an odd prime labeling is called an <i>odd prime graph</i>. We give some families of odd prime graphs and give some necessary conditions for a graph to be odd prime. Finally, we conjecture that every prime graph is odd prime graph.

scholarly journals The Metric Dimension and Local Metric Dimension of Relative Prime Graph

CAUCHY ◽  
2020 ◽  
Vol 6 (3) ◽  
pp. 149-161
Author(s):  
Inna Kuswandari ◽  
Fatmawati Fatmawati ◽  
Mohammad Imam Utoyo
Keyword(s):  
Prime Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Prime Graphs ◽  
Vertex Set ◽  
Integer Rings ◽  
Metric Dimensions ◽  
Relative Prime

This study aims to determine the value of metric dimensions and local metric dimensions of relative prime graphs formed from modulo  integer rings, namely . As a vertex set is  and  if  and  are relatively prime. By finding the pattern elements of resolving set and local resolving set, it can be shown the value of the metric dimension and the local metric dimension of graphs  are  and  respectively, where  is the number of vertices groups that formed multiple 2,3, … ,  and  is the cardinality of set . This research can be developed by determining the value of the fractional metric dimension, local fractional metric dimension and studying the advanced properties of graphs related to their forming rings.Key Words : metric dimension; modulo ; relative prime graph; resolving set; rings.

A Characterization of Finite Simple Groups by the Degrees of Vertices of Their Prime Graphs

2005 ◽  
Vol 12 (03) ◽  
pp. 431-442 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
A. R. Zokayi ◽  
M. R. Darafsheh
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Natural Numbers ◽  
Prime Graphs ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge, denoted by p~q, if there is an element in G of order pq. Assume [Formula: see text] with primes p1<p2<⋯<pkand natural numbers αi. For p∈π(G), let the degree of p be deg (p)=|{q∈π(G)|q~p}|, and D(G):=( deg (p1), deg (p2),…, deg (pk)). In this paper, we prove that if G is a finite group such that D(G)=D(M) and |G|=|M|, where M is one of the following simple groups: (1) sporadic simple groups, (2) alternating groups Apwith p and p-2 primes, (3) some simple groups of Lie type, then G≅M. Moreover, we show that if G is a finite group with OC (G)={29.39.5.7, 13}, then G≅S6(3) or O7(3), and finally, we show that if G is a finite group such that |G|=29.39.5.7.13 and D(G)=(3,2,2,1,0), then G≅S6(3) or O7(3).

scholarly journals On minimal prime extensions of a four-vertex graph in a prime graph

2004 ◽  
Vol 288 (1-3) ◽  
pp. 9-17 ◽  
Author(s):  
Andreas Brandstädt ◽  
Chính T. Hoàng ◽  
Jean-Marie Vanherpe
Keyword(s):  
Prime Graph ◽  
Vertex Graph ◽  
Minimal Prime

scholarly journals A Systematic Approach to Group Properties Using its Geometric Structure

2020 ◽  
Vol 13 (1) ◽  
pp. 84-95
Author(s):  
Muhammed Bello ◽  
Nor Muhainiah Mohd Ali ◽  
Nurfarah Zulkifli
Keyword(s):  
Prime Power ◽  
Prime Graph ◽  
Clique Number ◽  
Prime Power Order ◽  
Dihedral Groups ◽  
Cyclic Groups ◽  
Prime Graphs ◽  
Algebraic Properties ◽  
A Finite Group ◽  
The Relationship

The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic groups to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph’s vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral groups and less than five for cyclic groups. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.

scholarly journals On Minimal Prime Graphs and Posets

Order ◽  
2009 ◽  
Vol 26 (4) ◽  
pp. 357-375 ◽  
Author(s):  
Maurice Pouzet ◽  
Imed Zaguia
Keyword(s):  
Prime Graphs ◽  
Minimal Prime

scholarly journals A Systematic Approach to Group Properties Using its Geometric Structure

2020 ◽  
Vol 13 (1) ◽  
pp. 84-95
Author(s):  
Muhammed Bello ◽  
Nor Muhainiah Mohd Ali ◽  
Nurfarah Zulkifli
Keyword(s):  
Prime Power ◽  
Prime Graph ◽  
Clique Number ◽  
Prime Power Order ◽  
Dihedral Groups ◽  
Cyclic Groups ◽  
Prime Graphs ◽  
Algebraic Properties ◽  
A Finite Group ◽  
The Relationship

The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic groups to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph’s vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral groups and less than five for cyclic groups. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.

RECOGNIZING SOME FINITE SIMPLE GROUPS BY NONCOMMUTING GRAPH

2012 ◽  
Vol 11 (04) ◽  
pp. 1250077 ◽  
Author(s):  
M. KHEIRABADI ◽  
A. R. MOGHADDAMFAR
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Linear Groups ◽  
Nonabelian Group ◽  
Projective Special Linear Groups ◽  
Prime Graphs ◽  
Vertex Set ◽  
Trivial Center ◽  
Noncommuting Graph

Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the noncentral elements of G, and two distinct vertices x and y of ∇(G) are joined by an edge if and only if x and y do not commute as elements of G, i.e. [x, y] ≠ 1. The finite group L is said to be recognizable by noncommuting graph if, for every finite group G, ∇(G) ≅ ∇ (L) implies G ≅ L. In the present article, it is shown that the noncommuting graph of a group with trivial center can determine its prime graph. From this, the following theorem is derived. If two finite groups with trivial centers have isomorphic noncommuting graphs, then their prime graphs coincide. It is also proved that the projective special unitary groups U4(4), U4(8), U4(9), U4(11), U4(13), U4(16), U4(17) and the projective special linear groups L9(2), L16(2) are recognizable by noncommuting graph.

scholarly journals Finite groups with prime graphs of diameter 5

2020 ◽  
Vol 28 (3) ◽  
pp. 307-312
Author(s):  
Ilya B. Gorshkov ◽  
Andrey V. Kukharev
Keyword(s):  
Finite Groups ◽  
Prime Graph ◽  
Maximal Diameter ◽  
Prime Graphs

AbstractIn this paper we consider a prime graph of finite groups. In particular, we expect finite groups with prime graphs of maximal diameter.

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