**A Study on Energy of an Intuitionistic Fuzzy Directed
Graph**

** **

**Deepa G ^{1}*,^{ }Praba B^{ 2},
Chandrasekaran VM^{1}**

^{1}School of Advanced Sciences, VIT University,
Vellore-632014, Tamilnadu, India,

^{2}SSN College of Engineering, Kalavakkam, Chennai,
Tamilnadu, India,

*Corresponding Author E-mail:
**deepa.g@vit.ac.in , vmcsn@vit.ac.in, prabab@ssn.edu.in**

**ABSTRACT:**

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.

** **

**KEYWORDS:** Energy
of a graph, Fuzzy Graph, Intuitionistic Fuzzy Graph.

**INTRODUCTION:**

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 energy^{7,14,19,5,15}. Several criteria relate to energy such
as energy change due to edge addition, maximal energy, and equal energy has
been considered in^{1,19,5}. The energy of a graph has chemical
applications in^{11,12,14} and mathematical properties in^{13}.
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 B^{22}. 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
Kafmann^{18} in 1973, from the Zadeh’s fuzzy relations^{28-30}.
But Rosenfeld^{24} 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 Atanassov^{4}. In 1978,
Gutman^{10} 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 in^{6,21, 13}. Energy of
different graphs including regular^{16}, non-regular^{17},
circulant^{26} and random graphs^{27} is also under study.
Energy is defined for signed graphs in^{ 9} and for weighted graphs in^{25}.
The energy of graph is extended to the energy of fuzzy graph in^{2}.
The energy of fuzzy graph is extended to the energy of an intuitionistic fuzzy
graph in^{23}.

** **

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Received on 19.01.2016 Modified on 25.01.2016

Accepted on 22.02.2016 © RJPT All right reserved

*Research J. Pharm. and Tech. 9(2): Feb.,
2016; Page 190-195*

**DOI:** **10.5958/0974-360X.2016.00034.2**