A Study on Energy of an Intuitionistic Fuzzy Directed Graph
Deepa G1*, Praba B 2, Chandrasekaran VM1
1School of Advanced Sciences, VIT University, Vellore-632014, Tamilnadu, India,
2SSN College of Engineering, Kalavakkam, Chennai, Tamilnadu, India,
In this review article, we have discussed about energy of a graph and energy of a fuzzy graph. Further we presented some standard results on these graphs. Since most real world networks such as communication networks, data organization, computational devices, the flow of computation, web graphs etc are directed graphs, we have analyzed energy of an intuitionistic fuzzy directed graph through real time example.
The motivation about the study of graph energy comes from chemistry. In 1930 Erich Hückel proposed the Hückel molecular orbital theory. The basic problem in Hückel theory is to determine the eigen values and eigen vectors of the graph representing carbon atom connectivity of a given conjugated system. An interesting quantity in Hückel theory is the sum of the energies of all the electrons in a molecule, called total - electron energy7,14,19,5,15. Several criteria relate to energy such as energy change due to edge addition, maximal energy, and equal energy has been considered in1,19,5. The energy of a graph has chemical applications in11,12,14 and mathematical properties in13. The foundation for graph theory was laid in 1735 by Leonhard Euler when he solved the ‘Konigsberg bridges’ problem. Many real life problems can be represented by a graph. In computer science, graphs are used to represent networks of communications, data organization, computational devices, the flow of computation, etc.
The link structure of a website could be represented by a directed graph in which the vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B22. A similar approach can be taken to problems in travel, biology, computer chip design and many other fields. Hence graph theory is widely used in solving real time problems. But when the system is large and complex it is difficult to extract the exact information about the system using the classical graph theory. In such cases fuzzy graph is used to analyze the system.
The first definition of fuzzy graphs was proposed by Kafmann18 in 1973, from the Zadeh’s fuzzy relations28-30. But Rosenfeld24 introduced another elaborated definition including fuzzy vertex and fuzzy edges and several fuzzy analogs of graph theoretic concepts such as paths, cycles, connectedness and etc. The first definition of intuitionistic fuzzy graphs was proposed by Atanassov4. In 1978, Gutman10 introduced the concept of ‘graph energy’ as the sum of the absolute values of the eigen values of the adjacency matrix of the graph. Certain bounds on energy are discussed in6,21, 13. Energy of different graphs including regular16, non-regular17, circulant26 and random graphs27 is also under study. Energy is defined for signed graphs in 9 and for weighted graphs in25. The energy of graph is extended to the energy of fuzzy graph in2. The energy of fuzzy graph is extended to the energy of an intuitionistic fuzzy graph in23.
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Received on 19.01.2016 Modified on 25.01.2016
Accepted on 22.02.2016 © RJPT All right reserved