scholarly journals Some Properties of the Intersection Graph for Finite Commutative Principal Ideal Rings

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Emad Abu Osba ◽  
Salah Al-Addasi ◽  
Omar Abughneim
Keyword(s):  
Principal Ideal ◽  
Independence Number ◽  
Domination Number ◽  
Simple Graph ◽  
Intersection Graph ◽  
Principal Ideal Ring ◽  
Geodetic Number ◽  
Complement Graph ◽  
Nonzero Intersection ◽  
Work Done

Let R be a commutative finite principal ideal ring with unity, and let G(R) be the simple graph consisting of nontrivial proper ideals of R as vertices such that two vertices I and J are adjacent if they have nonzero intersection. In this paper we continue the work done by Abu Osba. We calculate the radius, eccentricity, domination number, independence number, geodetic number, and the hull number for this graph. We also determine when G(R) is chordal. Finally, we study some properties of the complement graph of G(R).

INTERSECTION GRAPH OF SUBMODULES OF A MODULE

2012 ◽  
Vol 11 (01) ◽  
pp. 1250019 ◽  
Author(s):  
S. AKBARI ◽  
H. A. TAVALLAEE ◽  
S. KHALASHI GHEZELAHMAD
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Clique Number ◽  
Simple Graph ◽  
Intersection Graph ◽  
Star Graph ◽  
Graph Theoretic ◽  
Nonzero Intersection ◽  
One To One

Let R be a ring with identity and M be a unitary left R-module. The intersection graph of an R-moduleM, denoted by G(M), is defined to be the undirected simple graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have nonzero intersection. We investigate the interplay between the module-theoretic properties of M and the graph-theoretic properties of G(M). We characterize all modules for which the intersection graph of submodules is connected. Also the diameter and the girth of G(M) are determined. We study the clique number and the chromatic number of G(M). Among other results, it is shown that if G(M) is a bipartite graph, then G(M) is a star graph.

The intersection graph of ideals of ℤm

2019 ◽  
Vol 11 (04) ◽  
pp. 1950037 ◽  
Author(s):  
S. Khojasteh
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Independence Number ◽  
Domination Number ◽  
Intersection Graph ◽  
Graph Structure ◽  
Chromatic Index ◽  
Vertex Set ◽  
Numerical Invariants ◽  
Positive Integers

Let [Formula: see text] be an integer, and let [Formula: see text] be the set of all non-zero proper ideals of [Formula: see text]. The intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a graph with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be an integer and [Formula: see text] be a [Formula: see text]-module. In this paper, we study a kind of graph structure of [Formula: see text], denoted by [Formula: see text]. It is the undirected graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Clearly, [Formula: see text]. Let [Formula: see text] and [Formula: see text], where [Formula: see text]’s are distinct primes, [Formula: see text]’s are positive integers, [Formula: see text]’s are non-negative integers, and [Formula: see text] for [Formula: see text] and let [Formula: see text], [Formula: see text]. The cardinality of [Formula: see text] is denoted by [Formula: see text]. Also, let [Formula: see text], [Formula: see text] and [Formula: see text] denote the independence number, the domination number and the set of all isolated vertices of [Formula: see text], respectively. We prove that [Formula: see text] and we show that if [Formula: see text] is not a null graph, then [Formula: see text] and [Formula: see text] We also compute some of its numerical invariants, namely maximum degree and chromatic index. Among other results, we determine all integer numbers [Formula: see text] and [Formula: see text] for which [Formula: see text] is Eulerian.

scholarly journals Complement of the generalized total graph of Zn

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 6103-6113 ◽  
Author(s):  
Chelvam Tamizh ◽  
M. Balamurugan
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Independence Number ◽  
Domination Number ◽  
Simple Graph ◽  
Total Graph ◽  
Vertex Set

Let R be a commutative ring with identity and H be a nonempty proper multiplicative prime subset of R. The generalized total graph of R is the (undirected) simple graph ?GTH(R) with all elements of R as the vertex set and two distinct vertices x and y are adjacent if and only if x + y ? H. The complement of the generalized total graph ?GTH(R) of R is the (undirected) simple graphwith vertex set R and two distinct vertices x and y are adjacent if and only if x + y < H. In this paper, we investigate certain domination properties of ?GTH(R). In particular, we obtain the domination number, independence number and a characterization for -sets in ?GTP(Zn) where P is a prime ideal of Zn. Further, we discuss properties like Eulerian, Hamiltonian, planarity, and toroidality of GTP(Zn).

On projective intersection graph of ideals of commutative rings

2019 ◽  
pp. 2150017
Author(s):  
V. Ramanathan
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Principal Ideal ◽  
Artinian Ring ◽  
Intersection Graph ◽  
Principal Ideal Ring ◽  
Ideal Formula ◽  
Ideal Ring ◽  
Projective Graph ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] the set of all nontrivial proper ideals of [Formula: see text]. The intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as the set [Formula: see text], and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring [Formula: see text], [Formula: see text] is a projective graph if and only if [Formula: see text] where [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] of nilpotency three and [Formula: see text] is a field. Furthermore, it is shown that for an Artinian ring [Formula: see text] [Formula: see text] if and only if [Formula: see text] where each [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] such that [Formula: see text]

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

scholarly journals Independence and domination on shogiboard graphs

2017 ◽  
Vol 4 (8) ◽  
pp. 25-37 ◽  
Author(s):  
Doug Chatham
Keyword(s):  
Independence Number ◽  
Domination Number ◽  
The Other ◽  
Independent Domination

Abstract Given a (symmetrically-moving) piece from a chesslike game, such as shogi, and an n×n board, we can form a graph with a vertex for each square and an edge between two vertices if the piece can move from one vertex to the other. We consider two pieces from shogi: the dragon king, which moves like a rook and king from chess, and the dragon horse, which moves like a bishop and rook from chess. We show that the independence number for the dragon kings graph equals the independence number for the queens graph. We show that the (independent) domination number of the dragon kings graph is n − 2 for 4 ≤ n ≤ 6 and n − 3 for n ≥ 7. For the dragon horses graph, we show that the independence number is 2n − 3 for n ≥ 5, the domination number is at most n−1 for n ≥ 4, and the independent domination number is at most n for n ≥ 5.

Roman domination and 2-independence in trees

2017 ◽  
Vol 09 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Nacéra Meddah ◽  
Mustapha Chellali
Keyword(s):  
Independence Number ◽  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Number Formula ◽  
Sum Formula ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] of [Formula: see text] for which [Formula: see text]. The weight of a RDF is the sum [Formula: see text], and the minimum weight of a RDF [Formula: see text] is the Roman domination number [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a [Formula: see text]-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text] The maximum cardinality of a [Formula: see text]-independent set of [Formula: see text] is the [Formula: see text]-independence number [Formula: see text] Both parameters are incomparable in general, however, we show that if [Formula: see text] is a tree, then [Formula: see text]. Moreover, all extremal trees attaining equality are characterized.

THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-II

2013 ◽  
Vol 12 (04) ◽  
pp. 1250199 ◽  
Author(s):  
T. ASIR ◽  
T. TAMIZH CHELVAM
Keyword(s):  
Commutative Ring ◽  
Independence Number ◽  
Clique Number ◽  
Commutative Rings ◽  
Intersection Graph ◽  
Vertex Connectivity ◽  
Total Graph ◽  
Graph Theoretic ◽  
Vertex Set ◽  
Finite Commutative Rings

The intersection graph ITΓ(R) of gamma sets in the total graph TΓ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about ITΓ(R) where R is a commutative Artin ring. In this paper, we continue our interest on ITΓ(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of ITΓ(R). Further, we focus on certain graph theoretic parameters of ITΓ(R) like the independence number, the clique number and the connectivity of ITΓ(R). Also, we obtain both vertex and edge chromatic numbers of ITΓ(R). In fact, it is proved that if R is a finite commutative ring, then χ(ITΓ(R)) = ω(ITΓ(R)). Having proved that ITΓ(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which ITΓ(R) is perfect. In this sequel, we characterize all commutative Artin rings for which ITΓ(R) is of class one (i.e. χ′(ITΓ(R)) = Δ(ITΓ(R))). Finally, it is proved that the vertex connectivity and edge connectivity of ITΓ(R) are equal to the degree of any vertex in ITΓ(R).

scholarly journals On k-rainbow domination in middle graphs

2021 ◽  
Author(s):  
Kijung Kim
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Matching Number ◽  
Domatic Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Middle Graph ◽  
Dominating Function

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.

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