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Fuzzy Magic Graphs-A Brief Study 1K.R International Journal of Pure and Applied Mathematics Volume 119 No. 15 2018, 1161-1170 ISSN: 1314-3395 (on-line version) url: http://www.acadpubl.eu/hub/ Special Issue http://www.acadpubl.eu/hub/ Fuzzy Magic Graphs-A Brief Study 1K.R. Sobha, 2S. Chandra Kumar and 3R.S. Sheeba 1Department of Mathematics, Sree Ayyappa College for Women, Chunkankadai, India. [email protected] 2Department of Mathematics, Scott Chirstian College, Nagercoil, India. [email protected] 3P.G. Department of Mathematics, Sree Ayyappa College for Women, Chunkankadai, India. [email protected] Abstract Graph labeling is a currently emerging area in the research of graph theory. A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. A graph is said to be a fuzzy labeling graph if it has fuzzy labeling. A fuzzy labeling graph is said to be a fuzzy magic graph if σ(u)+µ(uv)+σ(v) has a same value for all u,v ∈V which is denoted as m0(G).Every fuzzy magic graph is a fuzzy labeling graph ,but the converse is not true. The application of fuzzy magic graph is illustrated with suitable example. Key Words:Fuzzy graph, fuzzy labeling, fuzzy magic graph, wheel graph, fan graph. Subject Classification:TT1-AM010 1161 International Journal of Pure and Applied Mathematics Special Issue 1. Introduction Fuzzy set is a newly emerging mathematical frame work exemplify the phenomenon of uncertainly in real life tribulations[1]. A fuzzy set is defined mathematically by assigning a value to each possible individual in the universe of discourse, representing its grade of membership which corresponds to the degree to which that individually similar or compatible with the concept represented the fuzzy set . Based on Zadeh’s Fuzzy relation the definition of a fuzzy graph was introduced by Kauffmann in 1973.One of the first important papers on fuzzy graph theory was by Azriel Rosenfeld[4]. Rosenfeld introduce and examined such concepts as paths, connectedness , bridges , cut vertices , forests and trees [2].A fuzzy graph is the generalisation of the crisp graph. Therefore it is natural many properties are similar to crisp graph and also it deviates at many places [6]. The notation of magic graph was first introduced by Sunitha and Vijayakumar in 1964[3]. He defined a graph to be magic if it has an edge-labelling, within the range of real numbers, such that the sum of the labels around any vertex equals some constant, independent of the choice of vertex.Kotzig and Rosa defined a magic labelling to be a total in which the labels are the integers from one to 푉 퐺 + 퐸 퐺 1 .The sum of labels on an edge and its two endpoints is constant. In this paper, section 1 contain basic definitions and in section 2 , fuzzy magic labelling for some graphs.e.g. Wheel, fan , paw, helm, butterfly. 2. Preliminaries Definition 2.1 Let U and V be two sets . Then is said to be a fuzzy relation from U into V if is a fuzzy set of UxV. Definition 2.2 A fuzzy graph G=(, 휇) is a pair of functions : 푉 → 0,1 푎푛푑 휇: 푉 × 푉 → [0,1], where for all u,v ∈ 푉,we have 휇(푢, 푣) ≤ (푢)ʌ(푣). Definition 2.3 A labeling of a graph is an assignment of a values to the vertices and the edges of a graph. Definition 2.4 A graph G=(, 휇) is said to be a fuzzy labeling graph if : 푉 → 0,1 푎푛푑 휇: 푉 × 푉 → [0,1],is bijective such that the membership value of edges and vertices are distinct and 휇 푢, 푣 < 푢 ʌ(푣) for all 푢, 푣 ∈ 푉 1162 International Journal of Pure and Applied Mathematics Special Issue Example 2.1 Definition 2.5 A fuzzy labeling graph is said to be a fuzzy vertex magic graph if 푢 + 휇(푢, 푣) + (푣) has a same magic value for all 푢, 푣 ∈ 푉 which is denoted as 푚0 퐺 Definition 2.6 A fuzzy labeling graph is said to be a fuzzy edge magic graph if 푢 + 휇(푢, 푣) + (푣) has a same magic value for all 푢, 푣 ∈ 푉 which is denoted as 푀0 퐺 . Definition 2.7 A fuzzy labeling graph is said to be a fuzzy totally magic graph if and only if there exist a fuzzy labelling f which is both fuzzy edge magic with magic value M0 and fuzzy vertex magic with magic value m0.It is not required that M0=m0. Definition 2.8 Let G= (V,휇, ) be a fuzzy graph. Then the degree of a vertex v is defined by 푑(푣) 푢≠푣= µ(푢, 푣) The minimum degree of G is 훿 퐺 = ʌ{푑(푣)/푣 ∈ 푉} The maximum degree of G is ∆ 퐺 = ᴠ{푑(푣)/푣 ∈ 푉} Remark 2.1 Degree of any pair of vertices u,v in magic fuzzy graph always different from each other’s and sum of degrees of nodes must be equal to twice the membership values of all arcs. Proposition 2.1 In a fuzzy magic graph G, the order of G is always greater than or equal to the size of G . i.e., O(G)≥ 푆(퐺) In any fuzzy magic graph G, the following inequality holds: 훿(퐺) ≤ ∆(퐺) ≤ 푆(퐺) ≤ 푂(퐺) 1163 International Journal of Pure and Applied Mathematics Special Issue Proof Let G=(V,, 휇) be any fuzzy magic graph, 푉 ≠ ∅, where − fuzzy subset of V and 휇 − 푓푢푧푧푦 subset of VxV. Since 훿(퐺) and ∆(퐺) are minimum and maximum degree of G respectively. Clearly, 훿 퐺 ≤ ∆ 퐺 … … … … … . (1) We have, O(G)= 푥∈푉 (푥), the order of G S(G)= x≠y μ(x, y), the size of G By the definitions, 푥≠푦 휇(푥, 푦) ≥ 푉{푑(푣)/푣 ∈ 푉} i.e., S(G)≥ ∆ 퐺 … … … … … … (2) Also, in a fuzzy magic graph, the membership value of an edge is less than or equal to the minimum value of the edge connecting vertices. ∴ 푂 퐺 ≥ 푆 퐺 … … … … . (3) From (1),(2), and (3) we get, 훿 퐺 ≤ ∆ 퐺 ≤ 푆 퐺 ≤ 푂 퐺 . Example 2.2 Consider a fuzzy graph G=(V,휇, ).Let V={v1,v2,v3,v4},(v1)=0.4, (v1)=0.4 , (v2)=0.3 (v3)=0.2, (v4)=0.26, 휇(v1,v2)=0.04, 휇(v2,v3)=0.09, 휇(v3,v4)=0.16,휇(v2,v4)=0.02, 휇(v4,v1)=0.01,d(v1)=0.05,d(v2)=0.15, d(v3)=0.25,d(v4)=0.19,훿 퐺 = 0.05, ∆(G)=0.25,S(G)=0.32,O(G)=1.16 Thus,훿 퐺 ≤ ∆ 퐺 ≤ 푆 퐺 ≤ 푂 퐺 . 1164 International Journal of Pure and Applied Mathematics Special Issue Some Examples of Fuzzy Magic Graphs Butterfly Graph The butterfly graph is a plannar undirected graph with 5 vertices and 6 edges. Example 2.3 Pan graph The Pan graph is the graph obtained by joining a cycle graph to a singleton graph with a bridge. The special case of the 3-pan graph is sometimes known as the paw graph and the 4-pan graph as the banner graph. Example 2.4 Wheel Graph A Wheel graph is a graph formed by connecting a single vertex to all the vertices of a cycle. For n≥4 the wheel Wn is defined to be the graph K1+Cn-1 1165 International Journal of Pure and Applied Mathematics Special Issue Example 2.5 Fan Graph A Fan graph Fm,n is defined as the graph join Km+Pn, where Km is the empty graph on m vertices and Pn is the path graph on n vertices. The case m=1 corresponds to the usual fan graphs ,while m=2 corresponds to the double fan, etc., Example 2.6 Helm Graph The helm graph is the graph obtained from an wheel graph by adjoining a pendent edge at each vertex of the cycle. 1166 International Journal of Pure and Applied Mathematics Special Issue Example 2.7 Bull Graph The bull graph is a planar undirected graph with 5 vertices and 5 edges,in the form of a triangle with two disjoint pendant edges. Example 2.8 Result: For n≥1,a fuzzy Path Pn is consider as a fuzzy magic graph, where n is length of Pn. 3. Application of Fuzzy Vertex Magic Labeling A College wants to provide equal number of computer to its departments such that each computer is utilized either by one department or by two departments. Here the problem is to find the number of computers allocated to each department and also to find number of computers utilized by one department 1167 International Journal of Pure and Applied Mathematics Special Issue and number of computers used by two departments. Representing this situation as a graph by considering the departments as vertices and if two departments shares computers then there is an edge between the corresponding vertices. Considering the fuzzy vertex magic labeling to this graph we are able to get the solution needed. Problem A College wants to provide equal number of computers to its 5 departments Mathematics, English, Computer Science, Physics and Chemistry. In order to reduce the idle time of the computers the college allows few computers can be used by two departments. Suppose every department utilizing few computers which are utilized by other departments. Find the number of computers allotted to each department in percentage. Find the number of computers utilized by one department and by two departments in percentage.
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