Graph theory is one of the parts of current Mathshaving encountered a most noteworthy advancement lately. The theory of fuzzy graphs was created by in the year 1975. During a similarand have additionally presented different connectedness ideas in fuzzy graphs. Fuzzy set theory gives us not just an important and ground-breaking portrayal of measurement of vulnerabilities, yet an increasingly reasonable portrayal of dubious ideas communicated in natural languages. A few properties of unusual hubs, fringe hubs and focal hubs are gotten. The scientific implanting of ordinary set theory into fuzzy has become a natural marvel. Therefore the possibility of fluffiness is an improving one. This Research study analyzes the distancetotal conceptthat is a measurement, in a fuzzy graph is presented

R.G, BALAMURUGAN

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A STUDY ON DISTANCE OF FUZZY GRAPH THEORY Dr.R.G.BALAMURUGAN1,K.CHANDRAN2,M.MAHALAKSHMI3,A.SARANYA4 Department of Mathematics Dhanalakshmi Srinivasan College of  Arts and Science for Women (Autonomous) Perambalur  ABSTRACT Graph theory is one of the parts of current Mathshaving encountered a most noteworthy advancement lately. The theory of fuzzy graphs was created by in the year 1975. During a similar and have additionally presented different connectedness ideas in fuzzy graphs. Fuzzy set theory gives us not just an important and ground-breaking portrayal of measurement of vulnerabilities, yet an increasingly reasonable portrayal of dubious ideas communicated in natural languages. A few properties of unusual hubs, fringe hubs and focal hubs are gotten. The scientific implanting of ordinary set theory into fuzzy has become a natural marvel. Therefore the possibility of fluffiness is an improving one. This Research study analyzes the distancetotal conceptthat is a measurement, in a fuzzy graph is presented. Keywords: Fuzzy Graph Theory, Distance, eccentric, peripheral, central nodes, etc.  The idea of was started by L. . Fuzzy thoughts have been applied to other mathematical structures, for example, rings, gatherings, vector spaces, topologies and modules. Fuzzy set theory gives significant and ground-breaking portrayal of estimation of vulnerabilities, just as dubious ideas communicated in characteristic dialects.  Graph theory is a significant device to speak to numerous true issues. For instance, an informal community might be spoken to as a graph where vertices speak to accounts (establishments, people, and so on.) and edges speak to the connection between the records. On the off chance that the relations among the records are to be estimated as fortunate or unfortunate as indicated by the recurrence of contacts among the records, fuzzyness ought to be added to portrayal. This and numerous different issues propelled to characterize fuzzy graphs.   Fuzzy set theory is the mathematical improvement that holds extraordinary guarantee in turning into the meta language of equivocalness with regards to the frameworks worldview which been offered as a partner to the conventional science worldview. The theory of fuzzy sets can be considered as the marvel of uncertainty over all frameworks showing this property and its outcomes. Fuzzy set theory offers speculations of set theoretic ideas, for example, crossing point and association. In this manner it carries all out ideas into the dimensional domain. The thought of fuzzy set is a specialized instrument to mathematically get a handle on the utilization and the impact of unclear ideas in a way totally unique in relation to the manner in which old style mathematics is treating them.   II. SURVEYED STUDIES K. R. Bhutani (2003)presented the idea of complete fuzzy graphs and inferred that a total fuzzy graph has no cut nodes. The idea of solid curve in greatest crossing tree and its application in group examination and neural networks were concentrated by Sameera and Sunitha (2006).  P. Bhattacharya (1987) talked about certain properties of fuzzy graphs and presented the thought of erraticism and focus in fuzzy graphs.  A book by Frank Harary distributed in 1969 was tremendously famous and empowered mathematicians, physicists, electrical specialists and social researchers to converse with one another.  Rosenfeld (1979) considered on fuzzy sets and built up the structure of graphs, getting analogs of a few graph hypothetical ideas. He presented and analyzed such ideas as connectedness, ways, and extensions, groups, woodlands, cut vertices, and trees.  The idea of control in fuzzy graphs was examined by A. Somasundaram and S. Somasundaram. A. Somasundaram (1998) introduced the ideas of autonomous control, all out mastery, associated mastery and mastery in cartesian items and organization of fuzzy graphs. They additionally talked about mastery in fuzzy graph utilizing powerful edges.  Goudar et al. (2012) have acquired the geodetic number of line graph of any graph. Numerous limits on geodetic number as far as components of G and covering number of G is likewise gotten. Ekim et al. (2012) have demonstrated that  Of legitimate interim processed in. Polynomial Time algorithm is acquainted with register the geodetic number of square prickly plant graphs and a Polynomial Time algorithm to estimate the geodetic number of bipartite change graphs. It is additionally demonstrated that figuring the geodetic number on chordal graphs, bipartite graphs, co-graphs, split graphs and Ptolemaic graph is NP-hard.  Linda and Sunitha (2013) built up the ideas of m-geodetic spread, m-arched set, m-geodetic number, m-geodetic premise, m-raised hull, m-hull number and m-check hub. They have acquired adequate condition for a fuzzy graph to have its hub set as m-geodetic premise and furthermore examined m-fringe way, m-fringe vertex, and m-unusual vertex of fuzzy graph.  III. GRAPH THEORY APPLICATIONS Graph hypothetical ideas are comprehensively used to play out the different ideas and applications in various territories. Graph theory bargains various applications like image segmentation, data mining, clustering, group theory, coding theory, image capturing, and so on. So also utilizing graph ideas, displaying of network topologies should be possible. In software engineering, graphs are utilized to speak to networks of computational gadgets, correspondences, the progression of calculation and data association and so forth. Additionally in graph theory, circuits, paths and walks are having amazing applications in resource networking, database plan ideas and voyaging sales rep issue. In the comparable manner the most huge thought of graph shading is misused in resource allotment, booking. This prompts the upgrade of part of new algorithms and theorems which has been utilized in amazing applications. It is significant and educational when taking a gander at the spreading of disease, parasites and to get familiar with the effect of excursion that influence different species. Graph theory is likewise broadly utilized in science and safeguarding endeavors in which where the vertex play out the areas, where the genuine species exist and the edges play out the excursion in the middle of the districts. Graph hypothetical ideas are much of the time utilized in activities look into.  The most well-known and successful applications of networks in activities inquire about is booking and arranging of tremendous complex tasks. One of the best notable issues in tasks examine are "CPM (Critical Path Method) and PERT (Project Evaluation Review Technique)". In organic networks, graph theory are utilized as a demonstrating instrument that allows the misuse of most broad graphical invariants along these lines, it is probably going to decide optional Ribonucleic corrosive (RNA) plans numerically. This graphical invariants are deviations of the mastery number of a graph. By the examination the outcomes are worked out to clarify that the deviations of the control number which are utilized to recognize the trees that demonstrate its own structures and it is beyond the realm of imagination to expect to speak to RNA. IV. PRELIMINARIES Definition 1: A is a function’s pair σ: 𝑉 →  [0, 1] and 𝜇: 𝑉 × 𝑉 →  [0, 1], where for all 𝑢, 𝑣 𝑖𝑛 𝑉 we have 𝜇 (𝑢, 𝑣) ≤  𝜎 (𝑢) 𝛬 𝜎 (𝑣). Definition 2: Graph wherein the edges are unordered vertex pair is called an undirected graph. A graph wherein the edges are requested vertex pair is known as a coordinated graph. Thus if there is an edge from 𝑣𝑖to 𝑣𝑗  in G then 𝑣𝑖 , 𝑣𝑗  𝜖 𝐸. Definition 3: Let G: (σ, µ) be a fuzzy graph. The degree of a vertex u is 𝑑𝐺 𝑢 =𝑑 𝑢 =  𝑢 ≠ 𝑣 𝜇 𝑢, 𝑣 . Definition 4: A graph G is a couple of set (𝑉, 𝐸), meant by 𝐺 =  (𝑉, 𝐸),whereE is a set of edges& V is a set of . Each edge in E is a couple of in V. Each edge is related with a set comprising of it is possible that a couple of vertices called its end focuses. Definition 5: Let X be . A  A in X is defined as 𝐴 = {(𝑥, µ𝐴 (𝑥)) /𝑥 ∈  𝑋}which is described by a participation workµ𝐴 (𝑥): 𝑋 → [0, 1].Fuzzy set is an assortment of articles with evaluated enrollment for example having degrees of enrollment. Definition 6: Let 𝐺 =  (𝑉, 𝜎, µ) bewith |𝑉 |  =  𝑛 and 𝜇 ={𝑒1, 𝑒2, … . , 𝑒𝑚 ). If 𝑚𝑖 = 𝜇(𝑒𝑖)is the quality of the connection related with the ith edge, then  𝟐 𝒎𝒊𝟐 + 𝒏(𝒏 − 𝟏)𝒎𝒊=𝟏|𝑨|𝟐𝒏 ≤ 𝑬(𝑮) ≤  𝟐   𝒎𝒊𝟐𝒎𝒊=𝟏 𝒏. Definition 7: Let 𝐺 =  (𝑉, 𝜎, µ)beand A be its contiguousness matrix. The vitality of G is characterized as the whole of the total of the Eigen values of A. Definition 8: Let 𝐺 =  (𝑉, 𝜎, µ) be and A be its contiguousness matrix. The Eigen estimations of A are called Eigen estimations of G. The range of A is known as the range of G. It is meant by Spec G. Let 𝐺: (𝜎, 𝜇) is of the graph 𝐺: (𝑉, 𝐸), then the distance 𝑑 𝜎 𝑣𝑖 , 𝜎 𝑣𝑗  between two of its σ 𝑣𝑖 and σ 𝑣𝑗 is the length of shortest path between them, i.e. 𝑑 𝜎  𝑣 , 𝜎  𝑣𝑖  = 𝑀𝑖𝑛[ 𝜇(𝑢𝑖 , 𝑣𝑗 )𝑖 ,𝑗 ∈∧ . Definition 10:Nodes with least fuzzy unusualness are called fuzzy focal nodes or fuzzy outspread nodes, and nodes with greatest fuzzy unpredictability are called fuzzy peripheral nodes or fuzzy diametral nodes. V. FUZZY GRAPH STRONG SUM DISTANCE We attempted another thought for fuzzy distance two naming graph with appropriate clarification and the ideas of limit and inside independent on solid whole distance are presented. The distance 𝑑[𝜎(𝑣𝑖 ), 𝜎(𝑣𝑗 )] between two nodes 𝜎(𝑣𝑖 ) 𝑎𝑛𝑑 𝜎(𝑣𝑗) in a fuzzy graph as the length of the briefest way between them, i.e. 𝑑[𝜎(𝑣𝑖 ), 𝜎(𝑣𝑗)]  = 𝑀𝑖𝑛[𝛴𝑖, 𝑗 ∈  µ(𝑣𝑖 , 𝑣𝑗)]. Be that as it may, this definition doesn't fulfill the triangle imbalance.  Fuzzy graph  Let 𝐺: (𝑉, 𝜎, µ) be a connected. For any path 𝑃: 𝑢0 − 𝑢1 −𝑢2 − 𝑢3 − ⋯ . −𝑢𝑛 , length of P is defined as the weights total of the arcs in P i.e. 𝐿 𝑃 =  𝜇𝑛𝑖=1 (𝜇𝑖−1, 𝑢𝑖). If 𝑛 =  0, define 𝐿(𝑃)  =  0 and for 𝑛 ≥  1, 𝐿(𝑃)  >  0. for any two nodes 𝑢, 𝑣 𝑖𝑛 𝐺, 𝑙𝑒𝑡 𝑃 = { 𝑃𝑖 ∶ 𝑃𝑖 𝑖𝑠 𝑎 𝑢 –  𝑣 𝑝𝑎𝑡ℎ, 𝑖 =  1,2,3, …  }. thetotal between u and v is defined as 𝑑𝑠(𝑢, 𝑣)  =  𝑀𝑖𝑛 {𝐿(𝑃𝑖) ∶  𝑃𝑖 ∈ 𝑃, 𝑖 = 1,2,3, …  }. Theorem 1: In a 𝐺:  𝑉, 𝜎, 𝜇 , 𝑑𝑠 : 𝑉 × 𝑉 → 0,1 𝑖𝑠 𝑎 𝑚𝑒𝑡𝑟𝑖𝑐 𝑜𝑛 𝑉. 𝑖. 𝑒. ∀𝑢, 𝑣, 𝑤 ∈ 𝑉.  𝑑𝑠 𝑢, 𝑣 = 0 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑢 =𝑣  𝑑𝑠 𝑢, 𝑣 ≥ 0∀𝑢, 𝑣 ∈ 𝑉  𝑑𝑠 𝑢, 𝑤 ≤ 𝑑𝑠 𝑢, 𝑣 + 𝑑𝑠(𝑣, 𝑤)  𝑑𝑠 𝑢, 𝑣 = 𝑑𝑠(𝑣, 𝑢) Proof: Since reversal of a path fromis a path from and vice versa,𝑑𝑠 𝑢, 𝑣 = 𝑑𝑠(𝑣, 𝑢)Let 𝑃1 be a 𝑢 − 𝑣 path such that 𝑑𝑠(𝑢, 𝑣)  = 𝐿(𝑃1) 𝑎𝑛𝑑 𝑃2 be a 𝑣 −  𝑤 path such that 𝑑𝑠(𝑣, 𝑤)  =  𝐿(𝑃2). The path 𝑃1followed by 𝑃2is a u − w walk and since every walk contains one path, there exists a 𝑢 −  𝑤 path in G whose length is at most 𝑑𝑠(𝑢, 𝑣)  +𝑑𝑠(𝑣, 𝑤). 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑑𝑠(𝑢, 𝑤)  ≤𝑑𝑠(𝑢, 𝑣)  + 𝑑𝑠(𝑣, 𝑤). Remark.𝐼𝑓 µ(𝑢, 𝑣)  =  1 ∀ (𝑢, 𝑣)  ∈  µ ∗ Then𝑑𝑠(𝑢, 𝑣) is the length of the crisp graph shortest path. Theorem 2: If 𝐶∗ =  𝑁, 𝐴 indicate a fundamental crisp circular graph with 𝑁 =  𝐴 = 𝛼then exactly  2 cut nodes happened in the fuzzy distance two marking cycle graph𝐹(𝐶)𝜓 =  𝜎𝑁𝜓, 𝜇𝐴𝜓 . Proof: Let 𝐹(𝐶)𝜓 =  𝜎𝑁𝜓, 𝜇𝐴𝜓  is a fuzzy distance two labeling cycle with 𝐶𝜓 =(𝑁, 𝐴)as an underlying crisp circular graph where 𝑁 =  𝐴 = 𝛼. We realize that any node in a characterized a fuzzy cut node of that if its discharge gives the decrement in the estimation of solidarity of the connectedness for some another joined nodes. This suggests  is a fuzzy cut for that graph in the event that it fill in as a typical node of any two fuzzy extensions. Exactly1 bridges occurred in each fuzzy distance two labeling cycle graph 𝐹(𝐶)𝜓 = 𝜎𝑁𝜓, 𝜇𝐴𝜓 (i.e.) fuzzy distance two labeling cycle graph will have single weakest arc only. Let 𝜇𝐴𝜓(𝑝, 𝑞)denote the allocation of this single weakest arc of 𝐹(𝐶)𝜓 = 𝜎𝑁𝜓, 𝜇𝐴𝜓 and let 𝜎𝑁𝜓(𝑝)and 𝜎𝑁𝜓(𝑝) respectively be the assignment of the nodes p and q. Therefore excluding 𝜎𝑁𝜓  𝑝 &𝜇𝐴𝜓(𝑝, 𝑞)nodes from the graph 𝐹(𝐶)𝜓 =  𝜎𝑁𝜓, 𝜇𝐴𝜓 remaining all  2 nodes of 𝐹(𝐶)𝜓 =  𝜎𝑁𝜓, 𝜇𝐴𝜓 serve as a common node of two fuzzy bridges of the fuzzy distance two labeling cycle 𝐹(𝐶)𝜓 =  𝜎𝑁𝜓, 𝜇𝐴𝜓  . Therefore all these  2 nodes of will be cut nodes of 𝐹(𝐶)𝜓 =  𝜎𝑁𝜓, 𝜇𝐴𝜓 Hence exactly  2 cut nodes appeared in every fuzzy distance two labeling cycle 𝐹(𝐶)𝜓 =  𝜎𝑁𝜓, 𝜇𝐴𝜓 . Remarks: The node in a fuzzy distance two labeling cycle graph 𝐹(𝐶)𝜓 = 𝜎𝑁𝜓, 𝜇𝐴𝜓  is either a cut node or end node. Every fuzzy distance two labeling cycle graph 𝐹(𝐶)𝜓 =  𝜎𝑁𝜓, 𝜇𝐴𝜓 has exactly only one weakest arc, say𝜎𝑁𝜓  𝑝, 𝑞 . Which implies that 𝜎𝑁𝜓  𝑝 𝑎𝑛𝑑 𝜎𝑁𝜓(𝑞)are two end nodes. Hence exactly two end nodes obtained in every fuzzy distance two labeling cycle graph. Theorem 3: For every 2 strong neighbor’s v and u in a connected𝐺: (𝑉, 𝜎, 𝜇), |𝑒(𝑢)  −  𝑒(𝑣)|  ≤  1. Proof:Loss of generality 𝑒(𝑢)  ≥ 𝑒(𝑣). Let x be a . i.e. 𝑒 𝑢 = 𝑑𝑠𝑠 𝑢, 𝑥 ≤ 𝑑𝑠𝑠 𝑢, 𝑣 +𝑑𝑠𝑠(𝑣, 𝑥), by triangle inequality. Therefore 𝑒 𝑢 ≤ 𝑑𝑠𝑠 𝑢, 𝑣 + 𝑒 𝑣 , since 𝑒 𝑣 ≥𝑑𝑠𝑠 𝑣, 𝑥 . sincev and u are strong neighbors we have 𝑑𝑠𝑠(𝑢, 𝑣) ≤1.Therefore 𝑒 𝑣 ≤ 1 + 𝑒(𝑣) ⇒ 0 ≤  𝑒(𝑢)  −  𝑒(𝑣)  ≤  1. ∴ |𝑒 𝑢 −  𝑣 | ≤ 1. Theorem 4: 𝐺𝐴 𝜎, 𝜇 is an anti-fuzzy graph then the distance 𝑑(𝜎 𝑢 , 𝜎 𝑣 )between two of its vertices 𝜎 (𝑢) 𝑎𝑛𝑑 𝜎 (𝑣) is the path of shortest length between them, that is 𝑑 𝜎 𝑢 , 𝜎 𝑣  = min[ 𝜇(𝑢, 𝑣)𝑢 ,𝑣𝜖𝑉 ]  In Fig. 9, consider the path from 𝑢1 𝑡𝑜 𝑢3. Then there exists two path between(𝑢1) 𝑎𝑛𝑑 (𝑢3). They are  (𝑢1, 𝑢4),(𝑢4, 𝑢2),(𝑢2, 𝑢3)  (𝑢1, 𝑢4),(𝑢4, 𝑢3) The distance between the vertices 𝜎 𝑢1 , 𝑎𝑛𝑑 (𝑢3) Via path 2, is 𝜇 𝑢1, 𝑢4 + 𝜇 𝑢4, 𝑢2 + 𝜇(𝑢2, 𝑢3) = 0.3 + 0.6 + 0.6 = 1.5 Via path 1, is 𝜇 𝑢1, 𝑢4 + 𝜇 𝑢4, 𝑢3 = 0.3 + 0.5 = 0.8 Therefore, 𝑑( 𝜎(𝑢), 𝜎(𝑣) )  =  𝑚𝑖𝑛 [ 0.8 , 1.5 ] VI. CONCLUSION It is realized that fuzzy graph theory has various applications in present day science and engineering. We genuinely trust its wide going applications of and the interdisciplinary idea of , if appropriately mixed together could clear a route for a considerable development of . The idea of distance is significant as it speaks to the net stream between a given pair of nodes of a fuzzy graph. The idea of fuzzy focus, fuzzy self-centered graphs, and supplement of a fuzzy graph are additionally presented. The investigation of fuzzy graphs made right now a long way from being finished. Distance is a significant idea in the whole graph theory. Right now, certifiable exertion is made to sum up the idea of distance.  REFERENCES [1]. Zadeh, L.A (1965), Fuzzy Sets. Inform. Control 8, 338-353.  [2]. Bhutani, K.R., and Rosenfeld, A., “Fuzzy End Nodes in Fuzzy Graphs”, Information Sciences, Volume 152, June 2003, pp. 323-326. [3]. Bhattacharya. P, Some Remarks on fuzzy graphs, Pattern Recognition Letters, 6, 1987, pp. 297-302. [4]. Rosenfeld A. Fuzzy graphs. In: Zadeh LA, Fu KS, Shimura M, editors. Fuzzy sets and their applications to cognitive and decision processes. New York: Academic Press; 1979 pp. 77-95. [5]. Somasundaram .A, Somasundaram .S. 1998, Domination in Fuzzy Graphs – I, Pattern Recognition Letters, 19, pp-787-791. [6]. Frank Harary, Graph Theory, Narosa/ Addison Wesley, Indian Student Edition, 1988.  [7]. Mordeson, J.N. and Nair, P.S, “cycles and co-cycles of Fuzzy Graphs”, Inform. Sci. 90:39-49, 1996. [8]. M.delgado, J.LVerdegay and M.A.Vila, “On Fuzzy Tree Definition”, European J. operatios Res. 22(1985), 243-249. [9]. Tejaswini K.M, Venkanagouda M Goudar, Venkatesha and M.H.Muddebihal, On The Lict Geodetic Number of a Graph, International Journal of Mathematics and Computer Applications Research (IJMCAR), 2(3), 2012, pp.65-69. [10]. . Sameera K. and Sunitha M.S., “Strong Arcs and Maximum Spanning Trees in Fuzzy Graphs”, International Journal of Mathematical Sciences, 5(1) (2006), 17 – 20. [11]. 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A STUDY ON DISTANCE OF FUZZY GRAPH THEORY

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A STUDY ON DISTANCE OF FUZZY GRAPH THEORY

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