International Journal of Multidisciplinary Research and Publications ISSN (Online): 2581-6187 1 B. Gayathri and R. Gopi, “Some Properties of k-Mean Labeling,” International Journal of Multidisciplinary Research and Publications (IJMRAP), Volume 1, Issue 3, pp. 1-2, 2018. Some Properties of k-Mean Labeling B. Gayathri 1 , R. Gopi 2 1 Department of Mathematics, Periyar E.V.R. College, Tiruchirappalli, Tamilnadu, India-620023 2 Department of Mathematics, Simad Andavan Arts and Science College, Tiruchirappalli, Tamilnadu, India-620005 Email address: 1 maduraigayathri@gmail.com, 2 drrgmaths@gmail.com Abstract— A graph G with p vertices and q edges is called a k-mean labeling (k-ML) if there is an injective function f from the vertices of G to {0,1,2,...,k + q−1} such that when each edge uv is labeled with * () () ( ) 2 fu fv f uv then the resulting edge labels {k, k + 1,k + 2,...,k + q−1} are all distinct. A graph that admits k-mean labeling is called a k-mean graph(k-MG) Keywords— k-mean Labeling, k-mean graph.. I. INTRODUCTION All graphs in this paper are finite, simple and undirected. Terms not defined here are used in the sense of [5]. The symbols V(G) and E(G) will denote the vertex set and edge set of a graph. Labeled graphs serve as useful models for a broad range of applications [1]. A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. If the domain of the mapping is the set of vertices (or edges) then the labeling is called a vertex labeling (or an edge labeling). Graph labeling was first introduced in the late 1960s. Many studies in graph labeling refer to Rosas research in 1967[6]. Labeled graphs serve as useful models for a broad range of applications such as X-ray crystallography, radar, coding theory, astronomy, circuit design and communication network addressing. Particularly interesting applications of graph labeling can be found in [2]. S. Somasundaram and R. Ponraj, introduced the mean labeling. A graph G with p vertices and q edges is called a mean labeling if there is an injective function f from the vertices of G to f : V (G) →{0,1,...,q} such that when each edge uv is label with * () () ( ) 2 fu fv f uv then the resulting edge labels are distinct. A graph which admits mean labeling is called mean graph. Mean labeling of graphs was discussed in [4, 3]. B. Gayathri and R. Gopi, introduce the concept of A graph G with p vertices and q edges is called a k- mean labeling (k-ML) if there is an injective function f from the vertices of G to {0,1,2,...,k + q−1} such that when each edge uv is labeled with * () () ( ) 2 fu fv f uv then the resulting edge labels {k,k + 1,k + 2,...,k + q−1} are all distinct. A graph that admits k-mean labeling is called a k-mean graph(k-MG) In this paper, we have proved the some properties of k-mean labeling. II. MAIN RESULTS Theorem 2.1 Let G be a kmean graph with kmean labeling f. Let t be the number of edges whose one vertex label is even and the other is odd. Then 2 – 1 vVG q k d q v f v t where dv denotes the degree of a vertex v. Proof We know that odd is ) ( ) ( if 2 1 ) ( ) ( even is ) ( ) ( if 2 ) ( ) ( ) ( * y f x f y f x f y f x f y f x f xy f Now, vVG dv f v = * 2 2 xy E t f xy = 2 1 ... 1 2 t k k k q = 1 2 2 qq qk t = 2 1 2 2 qk qq t vVG dv f v t = 2qk + q(q – 1) = q(2k + q - 1) Corollary 2.2 If G is a kmean graph with kmean labeling f , then vVG dv f v q 2 . Proof From Theorem 2.1, dv f v = q(2k + q – 1) – t