scholarly journals Radio Number Associated with Zero Divisor Graph

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2187
Author(s):  
Ali N. A. Koam ◽  
Ali Ahmad ◽  
Azeem Haider

Radio antennas use different frequency bands of Electromagnetic (EM) Spectrum for switching signals in the forms of radio waves. Regulatory authorities issue a unique number (unique identifying call sign) to each radio center, that must be used in all transmissions. Each radio center propagates channels to the two nearer radio centers so they must use distinctive numbers to avoid interruption. The task of effectively apportioning channels to transmitters is known as the Channel Assignment (CA) problem. CA Problem is discussed under the topic of graph coloring by mathematicians. The radio number of a graph can be used in many parts of the field communication. In this paper, we determined the radio number of zero-divisor graphs Γ(Zp2×Zq2) for p,q prime numbers.

2011 ◽  
Vol 04 (03) ◽  
pp. 523-544 ◽  
Author(s):  
Laxman Saha ◽  
Pratima Panigrahi ◽  
Pawan Kumar

A number of graph coloring problems have their roots in a communication problem known as the channel assignment problem. The channel assignment problem is the problem of assigning channels (non-negative integers) to the stations in an optimal way such that interference is avoided, see Hale [4]. The radio coloring of a graph is a special type of channel assignment problem. Here we develop a technique to find an upper bound for radio number of an arbitrary graph and also we give a lower bound for the same. Applying these bounds we have obtained radio number of [Formula: see text], r ⩾ 3, for several values of n and r. Moreover for diameter 2 or 3 radio number of [Formula: see text] have been determined completely for all values of n and r.


2008 ◽  
Vol 308 (22) ◽  
pp. 5122-5135 ◽  
Author(s):  
Tongsuo Wu ◽  
Dancheng Lu

2011 ◽  
Vol 10 (04) ◽  
pp. 665-674
Author(s):  
LI CHEN ◽  
TONGSUO WU

Let p be a prime number. Let G = Γ(R) be a ring graph, i.e. the zero-divisor graph of a commutative ring R. For an induced subgraph H of G, let CG(H) = {z ∈ V(G) ∣N(z) = V(H)}. Assume that in the graph G there exists an induced subgraph H which is isomorphic to the complete graph Kp-1, a vertex c ∈ CG(H), and a vertex z such that d(c, z) = 3. In this paper, we characterize the finite commutative rings R whose graphs G = Γ(R) have this property (called condition (Kp)).


Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Osama Alkam ◽  
Emad Abu Osba

Let En be the ring of Eisenstein integers modulo n. In this paper we study the zero divisor graph Γ(En). We find the diameters and girths for such zero divisor graphs and characterize n for which the graph Γ(En) is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal.


2015 ◽  
Vol 14 (06) ◽  
pp. 1550079 ◽  
Author(s):  
M. J. Nikmehr ◽  
S. Khojasteh

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.


2021 ◽  
pp. 239-292
Author(s):  
David F. Anderson ◽  
T. Asir ◽  
Ayman Badawi ◽  
T. Tamizh Chelvam

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