Malaya Journal of Matematik, Vol. S, No. 1, 79-82, 2019 https://doi.org/10.26637/MJM0S01/0020 Fuzzy quotient-3 cordial labeling of star related graphs- Paper I P. Sumathi 1 * and J. Suresh Kumar 2 Abstract Let G( V, E ) be a simple, finite and planar graph of order p and size q. In this paper, the concept of Fuzzy Quotient-3 Cordial Labeling was introduced. Let σ : V (G) → [0, 1] be a function defined by σ (v)= r 10 , r ∈ Z 4 −{0}. For each edge uv define μ : E (G) → [0, 1] by μ (uv)= 1 10  3σ (u) σ (v)  where σ (u) ≤ σ (v). The function σ is called fuzzy quotient-3 cordial labeling of G if the number of vertices labeled with i and the number of vertices labeled with j differ by at most 1, the number of edges labeled with i and the number of edges labeled with j differ by at most 1 where i, j ∈  r 10 , r ∈ Z 4 −{0}  , i = j. The number of vertices having label i denotes v σ (i) and the number of edges having label i denotes e μ (i). Here it is proved that the Star graph and Star related graphs are Fuzzy Quotient-3 Cordial. Keywords Star, Cycle, Fuzzy quotient-3 cordial graph. AMS Subject Classification 05C15. 1 Department of Mathematics, C. Kandaswami Naidu College for Men, Chennai-600102, India. 2 Department of Mathematics, St. Thomas College of Arts and Science, Chennai-600107, India. *Corresponding author: 1 sumathipaul@yahoo.co.in; 2 jskumar.robo@gmail.com Article History: Received 21 December 2018; Accepted 11 February 2019 c 2019 MJM. Contents 1 Introduction ........................................ 79 2 Main Result ......................................... 80 3 Conclusion ......................................... 82 References ......................................... 82 1. Introduction Graphs considered here are finite and simple. Graph la- beling is used in several areas of science and technology like coding theory, astronomy, circuit design etc. The cordial la- beling concept was first introduced by cahit [2]. The quotient- 3 cordial labeling have been introduced by P. Sumathi, et al. found in [3–6]. They found some family of graphs are quotient-3 cordial. Motivated by these labelings we introduce fuzzy quotient-3 cordial labeling of graphs. If G admits fuzzy quotient-3 cordial labeling then G is called as fuzzy quotient-3 cordial graph. Definition 1.1. Let σ : V (G) → [0, 1] be a function defined by σ (v)= r 10 , r ∈ Z 4 −{0}. For each edge uv define μ : E (G) → [0, 1] by μ (uv)= 1 10  3σ (u) σ (v)  where σ (u) ≤ σ (v). The function σ is called fuzzy quotient-3 cordial labeling of G if the number of vertices labeled with i and the number of vertices labeled with j differ by at most 1, the number of edges labeled with i and the number of edges labeled with j differ by at most 1 where i, j ∈  r 10 , r ∈ Z 4 −{0}  , i = j. The number of vertices having label i denotes v σ (i) and the number of edges having label i denotes e μ (i). Definition 1.2. A complete bipartite graph K 1,n is said to be a star graph and it is denoted by G 1,n . Definition 1.3 ( C 3 (n) star graph). The graph obtained by attaching n pendent vertices in any one vertex of C 3 is called C 3 (n) star graph. Definition 1.4 ( C 3 (n, m) star graph). The graph obtained by attaching n pendent vertices in one vertex of C 3 , m pendent vertices in other one vertex of C 3 is called C 3 (n, m) star graph. Definition 1.5 ( C 3 (n, m, 1) star graph). The graph obtained by attaching n pendent vertices in one vertex of C 3 , m pen- dent vertices in other vertex of C 3 and 1 pendent vertices in remaining vertex of C 3 is called C 3 (n, m, 1) star graph.