On super (a,d)(a,d)-PhPh-antimagic total labeling of Stars

Let G=(V,E)G=(V,E) be a simple graph and HH be a subgraph of GG. GG admits an HH-covering, if every edge in E(G)E(G) belongs to at least one subgraph of GG that is isomorphic to HH. An (a,d)(a,d)-HH-antimagic total labeling of GG is bijection f:V(G)∪E(G)→{1,2,3,…,|V(G)|+|E(G)|}f:V(G)∪E(G)→{1,2,3,…,|V(G)|+|E(G)|} such that for all subgraphs H′H′ of GG isomorphic to HH, the H′H′ weights w(H′)=∑v∈V(H′)f(v)+∑e∈E(H′)f(e) constitute an arithmetic progression a,a+d,a+2d,…,a+(n−1)da,a+d,a+2d,…,a+(n−1)d where aa and dd are positive integers and nn is the number of subgraphs of GG isomorphic to HH. Additionally, the labeling ff is called a super (a,d)(a,d)-HH-antimagic total labeling if f(V(G))={1,2,3,…,|V(G)|}f(V(G))={1,2,3,…,|V(G)|}. In this paper we study super (a,d)(a,d)-PhPh- antimagic total labeling of the Star.