P2 (sic) H-super antimagic total labeling of comb product of graphs

Let L and H be two simple, nontrivial and undirected graphs. Let o be a vertex of H, the comb product between L and H, denoted by L (sic) H, is a graph obtained by taking one copy of L and vertical bar V(L)vertical bar copies of H and grafting the ith copy of H at the vertex o to the ith vertex of L. By definition of comb product of two graphs, we can say that V(L (sic) H) = {(a, v)vertical bar a is an element of V(L), v is an element of V(H)} and (a, v)(b, w) is an element of E(L (sic) H) whenever a = b and vw is an element of E(H), or ab is an element of E(L) and v = w = o. Let G = L (sic) H and P-2 (sic) H subset of G, the graph G is said to be an (a, d)-P-2 (sic) H-antimagic total graph if there exists a bijective function f : V(G) U E(G) -> {1, 2, ..., vertical bar V(G)vertical bar + vertical bar E(G)vertical bar} such that for all subgraphs isomorphic to P-2 (sic) H, the total P-2 (sic) H-weights W(P-2 (sic) H) = Sigma(v is an element of V(P2(sic)H))f(v)+ Sigma(e is an element of E(P2(sic)H))f(e) form an arithmetic sequence {a, a + d, a + 2d, ..., a + (n - 1)d}, where a and d are positive integers and n is the number of all subgraphs isomorphic to P2 (sic) H. An (a, d)-P-2 (sic) H-antimagic total labeling f is called super if the smallest labels appear in the vertices. In this paper, we study a super (a, d)-P-2 (sic) H-antimagic total labeling of G = L (sic) H when L = C-n. (C) 2019 Kalasalingam University. Production and Hosting by Elsevier B.V.