scholarly journals Prime Labeling of Jahangir Graphs

2018 ◽  
Vol 7 (4.10) ◽  
pp. 389
Author(s):  
Anantha Lakshmi. ◽  
Jayalakshmi. K ◽  
Madhavi T
Keyword(s):  
Prime Number ◽  
Prime Graph ◽  
Graph Operations ◽  
Prime Labeling ◽  
Jahangir Graph

The paper investigates prime labeling of Jahangir graph Jn,m   for n ≥ 2, m ≥ 3 provided that nm is even. We discuss prime labeling of some graph operations viz. Fusion, Switching and Duplication to prove that the Fusion of two vertices v1 and vk where k is odd in a Jahangir graph Jn,m results to prime graph provided that the product nm is even and is relatively prime to k. The Fusion of two vertices vnm + 1 and vk for any k in Jn, m is prime. The switching of vk in the cycle Cnm of the Jahangir graph Jn,m  is a prime graph provided that nm+1 is a prime number and the switching of vnm+1 in Jn, m is also a prime graph .Duplicating of vk, where k is odd integer and nm + 2 is relatively prime to k,k+2 in Jn,m is a prime graph.  

n-RECOGNITION BY PRIME GRAPH OF THE SIMPLE GROUP PSL(2,q)

2008 ◽  
Vol 07 (06) ◽  
pp. 735-748 ◽  
Author(s):  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Prime Number ◽  
Finite Groups ◽  
Prime Graph ◽  
Split Extension ◽  
A Finite Group

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, q are joined by an edge if there is an element in G of order pq. It is proved that if p > 11 and p ≢ 1 (mod 12), then PSL(2,p) is uniquely determined by its prime graph. Also it is proved that if p > 7 is a prime number and Γ(G) = Γ(PSL(2,p2)), then G ≅ PSL(2,p2) or G ≅ PSL(2,p2).2, the non-split extension of PSL(2,p2) by ℤ2. In this paper as the main result we determine finite groups G such that Γ(G) = Γ(PSL(2,q)), where q = pk. As a consequence of our results we prove that if q = pk, k > 1 is odd and p is an odd prime number, then PSL(2,q) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.

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.

On a conjecture about the quasirecognition by prime graph of some finite simple groups

2019 ◽  
Vol 18 (04) ◽  
pp. 1950070
Author(s):  
Ali Mahmoudifar
Keyword(s):  
Computer Program ◽  
Natural Number ◽  
Simple Group ◽  
Prime Number ◽  
Prime Power ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Number Formula ◽  
Finite Linear

It is proved that some finite simple groups are quasirecognizable by prime graph. In [A. Mahmoudifar and B. Khosravi, On quasirecognition by prime graph of the simple groups [Formula: see text] and [Formula: see text], J. Algebra Appl. 14(1) (2015) 12pp], the authors proved that if [Formula: see text] is a prime number and [Formula: see text], then there exists a natural number [Formula: see text] such that for all [Formula: see text], the simple group [Formula: see text] (where [Formula: see text] is a linear or unitary simple group) is quasirecognizable by prime graph. Also[Formula: see text] in that paper[Formula: see text] the author posed the following conjecture: Conjecture. For every prime power [Formula: see text] there exists a natural number [Formula: see text] such that for all [Formula: see text] the simple group [Formula: see text] is quasirecognizable by prime graph. In this paper [Formula: see text] as the main theorem we prove that if [Formula: see text] is a prime power and satisfies some especial conditions [Formula: see text] then there exists a number [Formula: see text] associated to [Formula: see text] such that for all [Formula: see text] the finite linear simple group [Formula: see text] is quasirecognizable by prime graph. Finally [Formula: see text] by a calculation via a computer program [Formula: see text] we conclude that the above conjecture is valid for the simple group [Formula: see text] where [Formula: see text] [Formula: see text] is an odd number and [Formula: see text].

CHARACTERIZING PSL2(r) BY ITS ORDER AND THE NUMBER OF ITS SYLOW r-SUBGROUPS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350065 ◽  
Author(s):  
ALIREZA KHALILI ASBOEI
Keyword(s):  
Prime Number ◽  
Prime Graph ◽  
Isolated Vertex ◽  
Mersenne Prime

We show that if r is a prime number that is not a Mersenne prime, then PSL2 (r) is determined up to isomorphism by its order and by the number of its Sylow r-subgroups. We then show that if r is a Mersenne prime other than 7, then PSL2 (r) is determined up to isomorphism by its order, the number of its Sylow r-subgroups, and the fact that r is an isolated vertex of the prime graph of the group.

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 Vertex Prime Valuation for Jahangir and Theta Graphs

2019 ◽  
Vol 8 (12) ◽  
pp. 5019-5021
Keyword(s):  
Disjoint Union ◽  
Graph Operations ◽  
Jahangir Graph

A sequence of instructions which can help to solve a problem is called an algorithm. The reason for composing an algorithm is to reduce the timespan and understanding the solution of problems in simple way. In this paper, vertex prime valuation of the Jahangir graph Jn,m for n ≥ 2, m ≥ 3 and generalized Theta graph θ (l1 , l 2 , l 3 , ..., ln) has been investigated by using algorithms .We discuss vertex prime valuation of some graph operations on both graphs viz. Fusion, Switching and Duplication, Disjoint union and Path union.

scholarly journals Prime labeling of line and splitting graph of brush graph

2021 ◽  
Vol 2106 (1) ◽  
pp. 012030
Author(s):  
F Fran ◽  
D R Putra ◽  
M Pasaribu
Keyword(s):  
Prime Graph ◽  
Bijective Function ◽  
Prime Labeling ◽  
Splitting Graph

Abstract A bijective function f from V(G) to {1,2,…, n} be a prime labeling of a graph G with n order if for every u, v ∈ V(G) such that e = uv ∈ E(G), f(u) and f(v) relatively prime. A prime graph is a graph which admits prime labeling. In this study, we investigate and conclude that the line and splitting graph of the brush graph is a prime graph.

On the prime graph of PSL(2, p) where p > 3 is a prime number

2007 ◽  
Vol 116 (4) ◽  
pp. 295-307 ◽  
Author(s):  
Bahman Khosravi ◽  
Behnam Khosravi ◽  
Behrooz Khosravi
Keyword(s):  
Prime Number ◽  
Prime Graph

QUASIRECOGNITION BY PRIME GRAPH OF SOME ORTHOGONAL GROUPS OVER THE BINARY FIELD

2012 ◽  
Vol 11 (03) ◽  
pp. 1250056 ◽  
Author(s):  
BEHROOZ KHOSRAVI ◽  
HOSSEIN MORADI
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Prime Graph ◽  
Composition Factor ◽  
Orthogonal Groups ◽  
Binary Field ◽  
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 G has an element of order pp′. Let D = Dn(2), where n ≥ 3 or 2Dn(2), where n ≥ 15. In this paper we prove that D is quasirecognizable by prime graph, i.e. every finite group G with Γ(G) = Γ(D), has a unique nonabelian composition factor which is isomorphic to D. Finally, we consider the quasirecognition by spectrum for these groups. Specially we prove that if p = 2n + 1 ≥ 17 is a prime number, then Dp(2) is recognizable by spectrum.

On Hyper-Zagreb Index of Three Graph Operations

2019 ◽  
Vol 10 (2) ◽  
pp. 301-309
Author(s):  
A. Bharali ◽  
Amitav Doley
Keyword(s):  
Zagreb Index ◽  
Graph Operations
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