On super H-Antimagicness of an Edge Comb Product of Graphs with Subgraph as a Terminal of its Amalgamation

Abstract

All graphs in this paper are simple, nite, and undirected graph. Let r be a edges of H. The edge comb product between L and H, denoted by LB H, is a graph obtained by taking one copy of L and jE(L)j copies of H and grafting the i-th copy of H at the edges r to the i-th edges of L, we call such a graph as an edge comb product of graph with subgraph as a terminal of its amalgamation, denoted by G = KBAmal(H; L H; n). The graph G is said to admits an (a; d)-H-antimagic total labeling if there exist a bijection f : V (G) [ E(G) ! f1; 2; : : : ; jV (G)j + jE(G)jg such that for all subgraphs isomorphic to H, the total H-weights W(H) = P v2V (H) f(v) + P f(e) form an arithmetic sequence fa; a + d; a + 2d; :::; a + (t 1)dg, where a and d are positive integers and t is e2E(H) the number of all subgraphs isomorphic to H. An (a; d)-H-antimagic total labeling f is called super if the smallest labels appear in the vertices. In this paper, we will study the super Hantimagicness of disjoint union of edge comb product of graphs with subgraph as a terminal of its amalgamation.