Ratio Mathematica Volume 46, 2023 Some results on range labeling J. Senthamizh Selvan* R. Jahir Hussain† Abstract Let G=( , ) be a graph with finite and simple . Let (G) and (G) be the vertex set and edge set of G respectively. A Range labeling of a graph G is an injective function. (G) →{1, 2. . . .} such that the edge labeling α∗ : (G) → {1, 2. . . ..} is defined by α[]= maximum value ( ) - minimum Value (). A graph which admits such labeling is called a range graph. In this paper the range labeling is introduced and range labeling for a some trees as for star tree, spider tree and Banana tree are calculated. Keywords: Graph; Labeling; Graceful labeling; Range labeling; Star trees; Spider trees; Banana trees. 2020 AMS subject classifications: 05C78 1 *Department of Mathematics (Jamal Mohamed College (Autonomous), Bharathidasan Univer- sity, Thiruchirappalli-620020, Tamil Nadu, India); senthamizh16@gmail.com †Department of Mathematics (Jamal Mohamed College (Autonomous), Bharathidasan Univer- sity, Thiruchirappalli-620020, Tamil Nadu, India); hssnjhr@yahoo.com 1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20, 2023. DOI: 10.23755/rm.v46i0.1080 ISSN: 1592-7415. eISSN: 2282-8214. ©J. Senthamizh Selvan et al. This paper is published under the CC-BY licence agreement. 236 J. Senthamizh Selvan and R. Jahir Hussain 1 Introduction Graph labeling is an assignment of set of integers to the set of vertices, edges or both based on certain conditions, In 1967 Rosa was first introduced the graph labeling. A graph labeling are useful family of mathematical models applied in many areas such as radar, missile guidance, radio frequency modulation and many more. A graph labeling is one topic in graph theory. So many kind of graph labeling among others: magic labeling, graceful labeling Afsana Ahmed munia [2014], o - Edge magic labeling, 1- Edge magic labeling, mean labeling, and etc. Every year and updated survey comes about various types of labeling by J. A. Gallian. From the survey, various types of labeling analyzed and introduced a new type of labeling called Range Labeling Moinin Al Aziz Md [2014]. In this paper are to proof that some trees namely Star tree, Spider tree, and Banana tree admit range labeling. This article assumed for all graphs are finite and simple. The graph G = (Ψ,τ) where Ψ(G) set of vertices and τ(G) set of edges. A labeling is a most one of the part of the graph theory, R.Uma [2012]. A labeling is the assignment of labels, traditionary defined by integers to edges or vertices or both of vertices and edges Moinin Al Aziz Md [2014]. The origin of labeling is Rosa by 1967. R. Jahir Hussain [2022] and Afsana Ahmed munia [2014] was First developed Range labeling in 2022. The article Range labeling apply for star trees, spider trees, Banana tress A. N.Mohamed [2013]. 2 Preliminary Definition 2.1. R.Uma [2012] A graceful labeling of a graph G is a vertex label- ing α : Ψ → [0,m] such that α is injective and induced mapping α(τ) = |α(Ψk) −α(Ψk+1)|, for every ΨkΨk+1 ∈ τ(G). Assigns different labels to different edges of G. The differences |α(Ψk)−α(Ψk+1)| is labeled weight of the edges ΨkΨk+1. A graph G is called graceful. Definition 2.2. Let G = (Ψ,τ) be a graph with n vertices. A bijection on α : Ψ → {1, 3, 6, 10, 15, ........n 2+n 2 } is called a range labeling if for each edge τ is distinct and τ is defined by α∗(τ) = Maximum value (Ψk, Ψk+1)− Minimum value (Ψk, Ψk+1). Definition 2.3. A tree for 1-internal vertex and k edges is called star S1,K that appear to be complete by bipartite graph K1,K. 237 Some results on range labeling Definition 2.4. A Spider tree with atmost one vertex of degree greater than 2. Definition 2.5. An (m,t)-Banana tree is a graph attained by attaching 1-edge of all m copies of an t-star graph for a 1-root vertex is different from each stars. 3 Main results Theorem 3.1. All star tree take a range labeling. Proof. Let G = (Ψ,τ), be a graph for Ψ1 is an interval vertex, 10 edges. Consider, α : Ψ →{1, 3, 6, 10, 15, ........n 2+n 2 } α∗(τ) = Maximum value (Ψk, Ψk+1)− Minimum value (Ψk, Ψk+1) α∗(τ1) = Maximum value (Ψ1, Ψ2)− Minimum value (Ψ1, Ψ2) if Ψ1 is a maximum value, Ψ2 is a minimum value. α∗(τ1) = (Ψ1 − Ψ2). = 1 is an integer. Suppose Ψ2 is a maximum value, Ψ1 is a minimum value. α∗(τ1) = (Ψ2 − Ψ1). = 1 is an integer. α∗(τ2) = Maximum value (Ψ1, Ψ3)− Minimum value (Ψ1, Ψ3) if Ψ1 is a maximum value, Ψ3 is a minimum value. α∗(τ2) = (Ψ1 − Ψ3). = 1 is an integer. Suppose Ψ3 is a maximum value, Ψ1 is a minimum value. α∗(τ2) = (Ψ3 − Ψ1). = 1 is an integer. α∗(τ9) = Maximum value (Ψ1, Ψ10)− Minimum value (Ψ1, Ψ10) if Ψ1 is a maximum value, Ψ10 is a minimum value. α∗(τ9) = (Ψ1 − Ψ10). = 1 is an integer. Suppose Ψ10 is a maximum value, Ψ1 is a minimum value. α∗(τ9) = (Ψ10 − Ψ1). = 1 is an integer. α∗(τ10) = Maximum value (Ψ1, Ψ11)− Minimum value (Ψ1, Ψ11) if Ψ1 is a maximum value, Ψ11 is a minimum value. α∗(τ10) = (Ψ1 − Ψ11). = 1 is an integer. Suppose Ψ11 is a maximum value, Ψ1 is a minimum value. α∗(τ10) = (Ψ11 − Ψ1). 238 J. Senthamizh Selvan and R. Jahir Hussain = 1 is an integer. Hence, Every star tree is a received range labeling Therefore, Any star tree is a Range graph. Example 3.1. 1 28 36 4555 66 3 6 10 15 21 27 35 4454 65 2 5 9 14 20 Fig.1. Range Labelling for star tree S10. Theorem 3.2. Every Spider tree take a range labeling. Proof. Let G = (Ψ,τ), be a graph. Let α : Ψ →{1, 3, 6, 10, 15, ........n 2+n 2 } A spider tree for atmost 1-node of degree greater than 2 and this node is said to be section node and is denoted by Ψ0. A stage of a spider graph is a path from the section node to edge of the tree. The edge is denoted by Ψ1, Ψ2, Ψ3. α∗(τ) = Maximum value (Ψk, Ψk+1)− Minimum value (Ψk, Ψk+1) α∗(τ1) = Maximum value (Ψ0, Ψ1)− Minimum value (Ψ0, Ψ1) if Ψ0 is a maximum value, Ψ1 is a minimum value. α∗(τ1) = (Ψ0 − Ψ1). = 1 is an integer. Suppose Ψ1 is a maximum value, Ψ0 is a minimum value. α∗(τ1) = (Ψ1 − Ψ0). = 1 is an integer. 239 Some results on range labeling α∗(τ2) = Maximum value (Ψ0, Ψ2)− Minimum value (Ψ0, Ψ2) if Ψ0 is a maximum value, Ψ2 is a minimum value. α∗(τ2) = (Ψ0 − Ψ2). = 1 is an integer. Suppose Ψ2 is a maximum value, Ψ0 is a minimum value. α∗(τ2) = (Ψ2 − Ψ0). = 1 is an integer. α∗(τ3) = Maximum value (Ψ0, Ψ3)− Minimum value (Ψ0, Ψ3) if Ψ0 is a maximum value, Ψ3 is a minimum value. α∗(τ3) = (Ψ0 − Ψ3). = 1 is an integer. Suppose Ψ3 is a maximum value, Ψ0 is a minimum value. α∗(τ3) = (Ψ3 − Ψ0). = 1 is an integer. So, All spider tree accepted range labeling Thus, Every spider tree is a range graph. Example 3.2. 1 10 6 3 9 5 2 Fig.2. Range Labelling for Spider Tree. 1015 6 1 3 75 2 3 Fig.3. Range Labelling for Spider Tree. Example 3.3. 240 J. Senthamizh Selvan and R. Jahir Hussain 6 3 36 1 136 10 15 105 28 91786621 120 5545 2 25 63 77 50 3 9 5 2021 121 34 45 99 15 Fig.1. Range Labelling for Banana tree B(3,5). Theorem 3.3. All Banana tree take a range labeling. Proof. Let G = (Ψ,τ), be a graph. Let α : Ψ →{1, 3, 6, 10, 15, ........n 2+n 2 } B(3, 5) is a Banana tree by attaching 1-edge of every 3 copies (Ψ1, Ψ2, Ψ3) of an 5−star (Ψ1., Ψ2., Ψ3.) graph for 1-root vertex is different from every stars. α∗(τ) = Maximum value (Ψk, Ψk+1)− Minimum value (Ψk, Ψk+1) α∗(τ1) = Maximum value (Ψ0, Ψ1)− Minimum value (Ψ0, Ψ1) if Ψ0 is a maximum value, Ψ1 is a minimum value. α∗(τ1) = (Ψ0 − Ψ1). = 1 is an integer. Suppose Ψ1 is a maximum value, Ψ0 is a minimum value. α∗(τ1) = (Ψ1 − Ψ0). = 1 is an integer. α∗(τ2) = Maximum value (Ψ0, Ψ2)− Minimum value (Ψ0, Ψ2) if Ψ2 is a maximum value, Ψ0 is a minimum value. α∗(τ2) = (Ψ2 − Ψ0). = 1 is an integer. Suppose Ψ0 is a maximum value, Ψ2 is a minimum value. α∗(τ2) = (Ψ0 − Ψ2). = 1 is an integer. α∗(τ3) = Maximum value (Ψ0, Ψ3)− Minimum value (Ψ0, Ψ3) 241 Some results on range labeling if Ψ0 is a maximum value, Ψ3 is a minimum value. α∗(τ3) = (Ψ0 − Ψ3). = 1 is an integer. Suppose Ψ3 is a maximum value, Ψ0 is a minimum value. α∗(τ3) = (Ψ3 − Ψ0). = 1 is an integer. α∗(τ4) = Maximum value (Ψ1, Ψ11)− Minimum value (Ψ1, Ψ11) if Ψ1 is a maximum value, Ψ11 is a minimum value. α∗(τ4) = (Ψ1 − Ψ11). = 1 is an integer. Suppose Ψ11 is a maximum value, Ψ1 is a minimum value. α∗(τ4) = (Ψ11 − Ψ1). = 1 is an integer. α∗(τ7) = Maximum value (Ψ1, Ψ14)− Minimum value (Ψ1, Ψ14) if Ψ1 is a maximum value, Ψ14 is a minimum value. α∗(τ7) = (Ψ1 − Ψ14). = 1 is an integer. Suppose Ψ14 is a maximum value, Ψ1 is a minimum value. α∗(τ7) = (Ψ14 − Ψ1). = 1 is an integer. α∗(τ8) = Maximum value (Ψ2, Ψ21)− Minimum value (Ψ2, Ψ21) if Ψ2 is a maximum value, Ψ21 is a minimum value. α∗(τ8) = (Ψ2 − Ψ21). = 1 is an integer. Suppose Ψ21 is a maximum value, Ψ2 is a minimum value. α∗(τ8) = (Ψ21 − Ψ2). = 1 is an integer. α∗(τ11) = Maximum value (Ψ2, Ψ24)− Minimum value (Ψ2, Ψ24) if Ψ2 is a maximum value, Ψ24 is a minimum value. α∗(τ11) = (Ψ2 − Ψ24). = 1 is an integer. Suppose Ψ24 is a maximum value, Ψ2 is a minimum value. α∗(τ11) = (Ψ24 − Ψ2). = 1 is an integer. α∗(τ12) = Maximum value (Ψ3, Ψ31)− Minimum value (Ψ3, Ψ31) if Ψ3 is a maximum value, Ψ31 is a minimum value. α∗(τ12) = (Ψ3 − Ψ31). = 1 is an integer. 242 J. Senthamizh Selvan and R. Jahir Hussain Suppose Ψ31 is a maximum value, Ψ3 is a minimum value. α∗(τ12) = (Ψ31 − Ψ3). = 1 is an integer. α∗(τ15) = Maximum value (Ψ3, Ψ34)− Minimum value (Ψ3, Ψ34) if Ψ3 is a maximum value, Ψ34 is a minimum value. α∗(τ15) = (Ψ3 − Ψ34). = 1 is an integer. Suppose Ψ34 is a maximum value, Ψ3 is a minimum value. α∗(τ15) = (Ψ34 − Ψ3). = 1 is an integer. Hence, Every Banana tree received range labeling. Therefore, All Banana tree is a range labeling. 4 Conclusions In this article discussed for some trees received range labeling, so this trees star tree, spider tree, banana tree is also a range graph. Further more Analysis for this labeling apply for some special graphs. References A. N.Mohamed. 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