scholarly journals On Super Mean Labeling for Total Graph of Path and Cycle

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Nur Inayah ◽  
I. Wayan Sudarsana ◽  
Selvy Musdalifah ◽  
Nurhasanah Daeng Mangesa
Keyword(s):  
Undirected Graph ◽  
Total Graph ◽  
Vertex Set ◽  
Multiple Edges

Let G(V,E) be a graph with the vertex set V and the edge set E, respectively. By a graph G=(V,E) we mean a finite undirected graph with neither loops nor multiple edges. The number of vertices of G is called order of G and it is denoted by p. Let G be a (p,q) graph. A super mean graph on G is an injection f:V→{1,2,3…,p+q} such that, for each edge e=uv in E labeled by f⁎e=fu+f(v)/2, the set fV∪{f⁎e:e∈E} forms 1,2,3,…,p+q. A graph which admits super mean labeling is called super mean graph. The total graph T(G) of G is the graph with the vertex set V∪E and two vertices are adjacent whenever they are either adjacent or incident in G. We have showed that graphs T(Pn) and TCn are super mean, where Pn is a path on n vertices and Cn is a cycle on n vertices.

Cycles and Connectivity in Graphs

1967 ◽  
Vol 19 ◽  
pp. 1319-1328 ◽  
Author(s):  
M. E. Watkins ◽  
D. M. Mesner
Keyword(s):  
Undirected Graph ◽  
Vertex Connectivity ◽  
Vertex Set ◽  
Multiple Edges

In this note, G will denote a finite undirected graph without multiple edges, and V = V(G) will denote its vertex set. The largest integer n for which G is n-vertex connected is the vertex-connectivity of G and will be denoted by λ = λ(G). One defines ζ to be the largest integer z not exceeding |V| such that for any set U ⊂ V with |U| = z, there is a cycle in G which contains U. The symbol i(U) will denote the component index of U. As a standard reference for this and other terminology, the authors recommend O. Ore (3).

Magic Valuations of Finite Graphs

1970 ◽  
Vol 13 (4) ◽  
pp. 451-461 ◽  
Author(s):  
Anton Kotzig ◽  
Alexander Rosa
Keyword(s):  
Undirected Graph ◽  
Complete Graphs ◽  
Finite Graphs ◽  
Vertex Set ◽  
Multiple Edges

The purpose of this paper is to investigate for graphs the existence of certain valuations which have some "magic" property. The question about the existence of such valuations arises from the investigation of another kind of valuations which are introduced in [1] and are related to cyclic decompositions of complete graphs into isomorphic subgraphs.Throughout this paper the word graph will mean a finite undirected graph without loops or multiple edges having at least one edge. By G(m, n) we denote a graph having m vertices and n edges, by V(G) and E(G) the vertex-set and the edge-set of G, respectively. Both vertices and edges are called the elements of the graph.

scholarly journals INTERSECTION GRAPH OF A MODULE

2013 ◽  
Vol 12 (05) ◽  
pp. 1250218 ◽  
Author(s):  
ERGÜN YARANERI
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Undirected Graph ◽  
Domination Number ◽  
Intersection Graph ◽  
Combinatorial Properties ◽  
Algebraic Properties ◽  
Vertex Set ◽  
Multiple Edges

Let V be a left R-module where R is a (not necessarily commutative) ring with unit. The intersection graph [Formula: see text] of proper R-submodules of V is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper R-submodules of V, and there is an edge between two distinct vertices U and W if and only if U ∩ W ≠ 0. We study these graphs to relate the combinatorial properties of [Formula: see text] to the algebraic properties of the R-module V. We study connectedness, domination, finiteness, coloring, and planarity for [Formula: see text]. For instance, we find the domination number of [Formula: see text]. We also find the chromatic number of [Formula: see text] in some cases. Furthermore, we study cycles in [Formula: see text], and complete subgraphs in [Formula: see text] determining the structure of V for which [Formula: see text] is planar.

DOMINATION IN THE TOTAL GRAPH ON ℤn

2011 ◽  
Vol 03 (04) ◽  
pp. 413-421 ◽  
Author(s):  
T. TAMIZH CHELVAM ◽  
T. ASIR
Keyword(s):  
Commutative Ring ◽  
Prime Divisor ◽  
Undirected Graph ◽  
Domination Number ◽  
Total Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Domination Numbers

For a commutative ring R, let Z(R) be its set of zero-divisors. The total graph of R, denoted by TΓ(R), is the undirected graph with vertex set R, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir studied about the domination in the total graph of a commutative ring R. In particular, it was proved that the domination number γ(TΓ(ℤn)) = p1 where p1 is the smallest prime divisor of n. In this paper, we characterize all the γ-sets in TΓ(ℤn). Also, we obtain the values of other domination parameters like independent, total and perfect domination numbers of the total graph on ℤn.

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

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).

A generalized ideal-based zero-divisor graph

2015 ◽  
Vol 14 (06) ◽  
pp. 1550079 ◽  
Author(s):  
M. J. Nikmehr ◽  
S. Khojasteh
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Proper Ideal ◽  
Graph Structure ◽  
Zero Divisor Graph ◽  
Vertex Set

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.

On an Algorithm for Ordering of Graphs

1971 ◽  
Vol 14 (2) ◽  
pp. 221-224 ◽  
Author(s):  
Milan Sekanina
Keyword(s):  
Undirected Graph ◽  
Multiple Edges

Let (G, ρ) be a finite connected (undirected) graph without loops and multiple edges. So x, y being two elements of G (vertices of the graph (G, ρ)), 〈x, y〉 ∊ ρ means that x and y are connected by an edge. Two vertices x, y ∊ G have the distance μ(x, y) equal to n, if n is the smallest number with the following property: there exists a sequence x0, x1, …, xn of vertices such that x0 = x, xn = y and 〈xi-1, Xi〉 ∊ ρ for i = 1, …, n. If x ∊ G, we put μ(x, x) = 0.

Proper connection of power graphs of finite groups

2020 ◽  
pp. 2150033
Author(s):  
Xuanlong Ma
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Undirected Graph ◽  
The Other ◽  
Connection Number ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Connection Number ◽  
A Finite Group

Let [Formula: see text] be a finite group. The power graph of [Formula: see text] is the undirected graph whose vertex set is [Formula: see text], and two distinct vertices are adjacent if one is a power of the other. The reduced power graph of [Formula: see text] is the subgraph of the power graph of [Formula: see text] obtained by deleting all edges [Formula: see text] with [Formula: see text], where [Formula: see text] and [Formula: see text] are two distinct elements of [Formula: see text]. In this paper, we determine the proper connection number of the reduced power graph of [Formula: see text]. As an application, we also determine the proper connection number of the power graph of [Formula: see text].

Bounds for the Genus of Generalized Total Graph of a Commutative Ring

2019 ◽  
Vol 26 (03) ◽  
pp. 519-528
Author(s):  
T. Asir ◽  
K. Mano
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Total Graph

Let R be a commutative ring with non-zero identity and I its proper ideal. The total graph of R with respect to I, denoted by T (ΓI (R)), is the undirected graph with all elements of R as vertices, and where distinct vertices x and y are adjacent if and only if [Formula: see text]. In this paper, some bounds for the genus of T(ΓI(R)) are obtained. We improve and generalize some results for the total graphs of commutative rings. In addition, we obtain an isomorphism relation between two ideal based total graphs.

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