* Corresponding Author. Email address: pathimathsloyola@gmail.com Characterization of Fuzzy Graphs into different categories using Arcs in Fuzzy Graphs T. Pathinathan 1* , J. Jesintha Rosline 1 (1) Department of Mathematics, Loyola College, Chennai – 34 Copyright 2014 © T. Pathinathan and J. Jesintha Rosline. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The relationships between different types of arcs in both regular and totally regular fuzzy graphs are analyzed in this paper. In these categories, the crisp graph obtained is nothing but a cycle. A characterization of totally regular fuzzy graph is obtained using arcs. Keywords: , ,  - arcs, fuzzy cycle, regular fuzzy graph, totally regular fuzzy graph, week totally regular fuzzy graph, strong totally regular fuzzy graph. 1 Introduction Fuzzy graphs were introduced by Rosenfeld [5], ten years after Zadeh’s seminal paper “Fuzzy Sets”[6]. Fuzzy graph theory is finding numerous application in the fields of information theory, neural networks, expert systems, cluster analysis, medical diagnosis, control theory, etc. Fuzzy modeling has become an efficient tool. Fuzzy models give more precision, flexibility and compatibility to the system when compared to classical models [6, 7]. Rosenfeld obtained the fuzzy analogues of several basic graph theoretic concepts like bridges, paths, cycles, trees and connectedness and established some of their properties [5]. Bhattacharya [8] had established some connectivity concepts regarding fuzzy cut nodes and fuzzy bridges. He has also introduced fuzzy groups and metric notion in fuzzy graphs. Bhutani [4] had studied automorphisms on fuzzy graphs and certain properties of complete fuzzy graphs. Bhattacharya and Suraweera had introduced an algorithm to find the connectivity of a pair of nodes in a fuzzy graph [3]. Nagoor Gani and Radha [1] defined regular and totally regular fuzzy graph. They stated some necessary and sufficient condition for these type of graphs. Sunil Mathew and Sunitha defined different types of arcs in fuzzy graphs and using them classified fuzzy graphs [2]. In this paper, a comparative study is made between regular and totally regular fuzzy graphs with reference to different types of arcs in fuzzy graphs. Also a necessary condition for a graph to be regular or totally regular is formulated in terms of arcs. Journal of Fuzzy Set Valued Analysis 2014 (2014) 1-6 Available online at www.ispacs.com/jfsva Volume 2014, Year 2014 Article ID jfsva-00222, 6 Pages doi:10.5899/2014/jfsva-00222 Research Article