F3086017618

Super (a,1)-Tree-Antimagicness of Sun Graphs
Muhammad Awais Umar¹, Mujtaba Hussain², Basharat Rehman Ali³, Muhammad Numan^4
1Muhammad Awais Umar, Abdus Salam School of Mathematical Sciences, G C University, Lahore, Pakistan.
²Mujtaba Hussain, Department of Mathematics, COMSATS Institute of Information Technology, Lahore, Pakistan.
³Basharat Rehman Ali, Abdus Salam School of Mathematical Sciences, G C University, Lahore, Pakistan.
⁴Muhammad Numan, Department of Mathematics, COMSATS Institute of Information Technology, Attock, Pakistan.
Manuscript received on December 21, 2018. | Revised Manuscript received on December 28, 2018. | Manuscript published on January 05, 2018. | PP: 1-4 | Volume-7 Issue-6, January 2018. | Retrieval Number: F3086017618/2018©BEIESP
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©The Authors. Published By: Blue Eyes Intelligence Engineering and Sciences Publication (BEIESP). This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/

Abstract: Let G = (V,E) be a finite simple graph with |V(G)| vertices and | E(G)| edges. An edge-covering of G is a family of subgraphs H H Ht , , , 1 2  such that each edge of E(G) belongs to at least one of the subgraphs Hi , i =1,2,,t . If every subgraph Hi is isomorphic to a given graph H , then the graph G admits an H -covering. A graph G admitting H covering is called an (a,d)- H -antimagic if there is a bijection f :V E {1,2,,|V(G)|  | E(G)|} such that for each subgraph H of G isomorphic to H , the sum of labels of all the edges and vertices belonged to H constitutes an arithmetic progression with the initial term a and the common difference d . For f (V) ={1,2,3,,|V(G)|} , the graph G is said to be super (a,d) – H -antimagic and for d = 0 it is called H -supermagic. In this paper, we investigate the existence of super (a,1)- 3 S -antimagic labeling of Sun graphs n SG , its uniform subdivision SG (r) n , disjoint union of sun graphs and its uniform subdivision denoted by mSGn and mSGn(r) respectively, where r, m ≥ 1.
Keywords: Sun graph n SG , uniform subdivided Sun graph SG (r) n , su- per (a, 1)-S3-antimagic, super (a, 1)-S3(r)-antimagic, disjoint union of Sun graph mSGn and its uniform subdivision mSGn(r). MR (2010) Subject Classification: 05C78, 05C70.