Informatics Publishing Limited

M. Ramachandran ¹, N. Parvathi ¹

Affiliations
1 Mathematics Department, S.R.M. University, Kattankulathur, Chennai-603 203, Tamil Nadu, IN

Abstract

In 1958 - Claude Berge introduced the domination number of a graph which is used to protect the single vertices. But in 2011 Duygu Vargor and Pinar Dundar initiated the medium domination number of a graph which is utilized to protect the pairs of vertices in a graph.

In a graph every vertices u, v ∈ V should be privileged and it is essential to scrutinize how many vertices are proficient of dominating both of u and v. We compute the total number of vertices that dominates all pairs of vertices and evaluate the average of this value and call it "the medium domination number" of graph. The medium domination number of G is the minimum cardinality among all the medium domination sets of G.

We prove the main result by two-dimensional induction method. First we are manipulative the medium domination number of J_1,3. Then we are calculating the medium domination number of J_m+1,3 and J_m,n+1. Finally we are getting the medium domination number of J_m,n. By using this method we can proficient to observe how many pairs of vertices are dominates in the Jahangir graph J_m,n.

In graph theory, there are many stability parameters such as the connectivity number, the edge-connectivity number, the independence number, the vertex domination number and the domination number. In this paper, we obtained the bound of the medium domination number of Jahangir graph J_m,n.

Keywords

Domination Number, Jahangir Graph, Medium Domination Number, TDV.

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G. Hemalatha ¹, N. Parvathi ²

Affiliations
1 Department of Mathematics, Odaiyappa College of Engineering and Technology, PTR Palanivelrajan Nagar, Theni, IN
2 Department of Mathematics, SRM University, Kattankulathur, Chennai, IN

Abstract

Let G=(V,E ) be a simple graph on a vertex set V. In a Graph G, A set D⊆V is a dominating set of G if every vertex in V-D is adjacent to some vertex in D. A dominating set D of G is minimal if for any vertex v∈D, D-{v} is not a dominating set of G. The domination number of a graph G, denoted by γ(G), is the minimum size of a dominating set of vertices in G. The domination subdivision number of a graph G is the minimum number of edges that must subdivided in order to increase the domination number of a graph and it is denoted by sdγ(G). A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v∈V is adjacent to an element of S. The total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. Total domination subdivision number denoted by sdγt is the minimum number of edges that must be subdivided to increase the total domination number. In this paper the domination subdivision number for some known graphs are investigated. In this paper the domination subdivision number for some known graphs are investigated.

Keywords

Dominating Set, Domination Subdivision Number, Cartesian Product, Total Domination Number, Total Domination Subdivision Number Cartesian Product.

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A. Vimala Rani ¹, N. Parvathi ¹

Affiliations
1 Department of Mathematics, Faculty of Science and Humanities, SRM University, Kattankulathur-603 203 Tamil Nadu, IN

Abstract

Objectives: In this study, Radio Coloring is used to color the graphs. The objective of this article is to analyze the bounds of Line, Middle, Total and Central graphs of Sunlet graph. Methods/Analysis: Combinatorics is an important part of discrete mathematics that solves counting problems without actually enumerating all possible cases. Combinatorics has wide applications in Computer Science, especially in coding theory, analysis of algorithms and others. An equation that expresses an, the general term of the sequence {an} is called a recurrence relation. Using the generating function of a sequence and few coloring techniques we prove the results. Findings: The Problem of finding radio coloring with small or optimal k arises in the concept of radio frequency assignment. The radio chromatic score rs(G)17 of a radio coloring is the number of used colors. The number of colors used in a radio coloring with the minimum score is the radio chromatic rn(G) of G. The radio chromatic number of Sunlet graph Sn is 10 4 if n=3i, i=1,2,... 5 if n=3i+1, i=1,2,... 6 if n=3i+2, i=1,2,... and is improved to the radio chromatic number of Sunlet graph Sn is 5 if n is congruent to 1mod 3 4 if otherwise In this paper we improve the radio chromatic number of Sunlet graph Sn and obtain the radio number of Line, Middle, Total and Central graphs of Sunlet graph. Radio Coloring has wide range of significance because radio coloring has its applications in communication theory. The paper contributes the researches in the field of computer science and combinatorics. Applications: Radio coloring is of great significance because the frequency assignment problem is modeled as a graph coloring problem assuming transmitters as vertices and interference as adjacencies between two vertices.

Keywords

Sunlet Graph, Line Graph, Middle Graph, Radio Number, Total Graph and Central Graph.

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S. Dhanalakshmi ¹, N. Parvathi ¹

Affiliations
1 Department of Mathematics, SRM University, Chennai – 603203, Tamil Nadu, IN

Abstract

Background/Objectives: Rough set theory proposed a new mathematical approach to vagueness or imperfect knowledge. It is the learning of approximations of concepts represented by lower and upper approximations which is being attracted by many researchers. The current study is a combination of rough sets approximation and graph labeling. Methods/Analysis: Many researchers have studied prime graph. Here we combine rough approximations with prime labeling under the name of H-prime labeling on graph G. Findings: The current work is to prove that the induced sub graph obtained by the upper approximation of any sub graph H of a friendship graph F_n, bistar graph B_n,n and splitting graph of a star graph 'S' ' graph admits prime labeling. The various applications between rough sets and graph labeling are chemical classification, decision analysis, knowledge acquisition, machine learning, job assignment etc.

Keywords

Bistar Graph, Friendship Graph, Lower and Upper Approximation, Prime Labeling, Rough Set, Splitting Graph.

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