IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 10, Issue 5 Ver. VI (Sep-Oct. 2014), PP 05-09 www.iosrjournals.org www.iosrjournals.org 5 | Page On Graph of a Finite Group R.A.Muneshwar 1 , K.L.Bondar 2 1 Department of Mathematics, NES Science College, SRTMU Nanded, India – 431602 2 Department of Mathematics, NES Science College, SRTMU Nanded, India – 431602 Abstract: In this paper we introduced a new concept of graph of any finite group and we obtained graphs of some finite groups. Moreover some results on this concept are proved. Keywords: Group, Abelian group, Cyclic group, Graph, Degree of a graph. I. Introduction: The origin of graph theory started with the problem of Koinsberg bridge, in 1735. This problem lead to the concept of Eulerian graph. Euler studied the problem of Koinsberg bridge and constructed a structure to solve the problem called Eulerian graph. In 1840, A.F Mobius gave the idea of complete graph and bipartite graph and Kuratowski proved that they are planar by means of recreational problems. The concept of tree was implemented by Gustav Kirchhoff in 1845, and he employed graph theoretical ideas in the calculation of currents in electrical networks or circuits. In 1852, Thomas Gutherie found the famous four color problem. Then in 1856, Thomas. P. Kirkman and William R.Hamilton studied cycles on polyhydra and invented the concept called Hamiltonian graph by studying trips that visited certain sites exactly once. In 1913, H.Dudeney mentioned a puzzle problem. Eventhough the four color problem was invented it was solved only after a century by Kenneth Appel and Wolfgang Haken. This time is considered as the birth of Graph Theory [1]. Caley studied particular analytical forms from differential calculus to study the trees. This had many implications in theoretical chemistry. This lead to the invention of enumerative graph theory. Any how the term “Graph” was introduced by Sylvester in 1878 where he drew an analogy between “Quantic invariants” and covariants of algebra and molecular diagrams. In 1941, Ramsey worked on colorations which lead to the identification of another branch of graph theory called extremel graph theory. In 1969, the four color problem was solved using computers by Heinrich. The study of asymptotic graph connectivity gave rise to random graph theory. Graph theory is rapidly moving into the mainstream of mathematics mainly because of its applications in diverse fields which include biochemistry, electrical, engineering (communications networks and coding theory), computer science (algorithms and computations) and operations research (scheduling). The powerful combinatorial methods found in graph theory have also been used to prove significant and well-known results in a variety of areas in mathematics itself. An application of matching in graph theory shows that there is a common set of left and right coset representatives of a subgroup in a finite group [1]. Up to this point, we have been looking at a group as a collection of elements that satisfy some conditions. Because graph has wide range of application in various fields, this motivates us to convert group into graph and make it applicable into various field. In this paper we try to bring very different way of representing the group, using the graph associated with the group rather than the algebraic structure of group. This paper is meant as an introduction and overview of some nice ideas from group theory by using graph theory. We convert group into graph and try to study various properties of group by using graph theory [1]. II. Some Basic Definitions: Following definitions are comes from references [2], [3], [4], [5], [6], [7], [8]. Definition 2.1 (Group): A nonempty set G with a binary operation . is called as a group if the following axioms hold: (i) a(bc) = (ab)c for all a,b,c G (ii) There exists e in G such that ea = ae = a ; G (iii) For every a G there exists a' G such that a' a =a a'= e. Definition 2.2 (Abelian group): A group G in which all elements satisfies commutative law is called as a abelian group. Definition 2.3 (Cyclic group): A group G is said to be cyclic if G = [a] = {x=a n | n Z }, for some a G.The most important examples of cyclic groups are the additive group Z of integers and the additive groups Z/(n) of integers modulo n. In fact, these are the only cyclic groups up to isomorphism.