International Journal of Mathematics And its Applications Volume 4, Issue 4 (2016), 187–191. (Special Issue) ISSN: 2347-1557 Available Online: http://ijmaa.in/ A p p l i c a t i o n s • I S S N : 2 3 4 7 - 1 5 5 7 • I n t e r n a t i o n a l J o u r n a l o f M a t h e m a t i c s A n d i t s International Journal of Mathematics And its Applications Harmonious Coloring of Middle and Central Graph of Some Special Graphs Research Article * J.Arockia Aruldoss 1 and S.Margaret Mary 1 1 Department of Mathematics, St. Joseph’s College of Arts & Science, Manjakuppam, Cuddalore (Tamil Nadu), India. Abstract: Let G(V,E) be an undirected graph. The Harmonious coloring of a graph G is a proper vertex coloring in which each pair of colors appears on at most one pair of adjacent vertices. The Harmonious chromatic number is minimum number of colors needed for the harmonious coloring of G. In this paper we investigate the harmonious chromatic number of middle and central graph of some special graphs. Keywords: Proper coloring, Middle graph, Central graph, Tadpole graph Web graphs. c  JS Publication. 1. Introduction In this paper, we have taken the graphs to be finite, undirected graphs. Every graph has a harmonious coloring, since it suffices to assign every vertex a distinct color. There trivially exist graphs G with χH(G) >χ(G). Then Akhalak Masuri, R.S.Chandel Vijay Gupta published a paper On Harmonious Coloring of M[Yn] and C[Yn]. In this paper we discuss about the harmonious chromatic number of central and middle graph of tadpole graph and web graph. 2. Preliminaries Definition 2.1 (Proper Coloring). A graph G having no two adjacent vertices receive the same color is said to be proper coloring. Definition 2.2 (Middle Graph). The M iddle graph of G denoted by M(G). The Vertex set of M (G) is V (G) ∪ E(G) in which two elements are adjacent in M(G) if the following conditions holds. (1) x, y ∈ E(G) and x, y are adjacent in G. (2) x ∈ V (G), y ∈ E(G) and they are incident in G Or A graph G is obtained by subdividing each edge of G exactly once and join all the newly middle vertices of adjacent edges of G is called the Middle graph. Definition 2.3 (Central Graph). The central graph of G, denoted by C(G) is obtained by subdividing each edge of G exactly once and joining all the non adjacent vertices G in C(G). * Proceedings : National Conference on Recent Trends in Applied Mathematics held on 22 & 23.07.2016, organized by Department of Mathematics, St. Joseph’s College of Arts & Science, Manjakuppam, Cuddalore (Tamil Nadu), India.