IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 10, Issue 6 Ver. II (Nov - Dec. 2014), PP 55-58 www.iosrjournals.org www.iosrjournals.org 55 | Page Some Properties of Graph of a Finite Group R.A.Muneshwar 1 , S.S.Agrawal 2 , S.G.Jakkewad 3 , 1 Department of Mathematics, NES Science College, SRTMU Nanded, India – 431602 2 Department of Mathematics, NES Science College, SRTMU Nanded, India – 431602 3 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 vertex, 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.[1] Muneshwar R. A. and Bondar K.L. in [2] introduced the concept of graph of a finite group. Some properties of graph of finite group are proved. In this paper some more properties with example are discussed. II. Preliminary Notes: Following definitions are comes from references [3], [4], [5], [6], [7], [8] [9]. 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 an 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. Definition 2.4 (Subgroup): Let (G, .) be a group and H be a subset of G. Then H is called a subgroup of G, if H is a group relative to the binary operation in G and it is denoted by H ≤G. Definition 2.5 (Center of a group): The center of a group G, written as Z(G), is the set of those elements in G that commute with every clement in G. That is Z(G) = (a ∈ G | ax = xa ∀ ∈ G }. Definition 2.6 (Centralizer of an element): Let g ∈ G be any element of group G then centralizer of an element is written as C(g), is the set of those elements in G that commute with element g . i.e.C(g)={a ∈ G | ag = ga }. Definition 2.7 (Centralizer of a subgroup): Let H be any subgroup of G then centralizer of a subgroup is written as C(H), is the set of those elements in G that commute with all elements of subgroup H. i.e. C(H) = {a ∈ G | ah = ha , ∀ ℎ∈ H }.