Super (a,d)-$H$- antimagic total covering of connected amalgamation of fan graph

Abstract

Graph $G=(V,E)$ is a finite, simple and undirected. Graph $G$ have $H'$ covering, if every edge in $E(G)$ belongs to at least one subgraph of $G$ isomorphic to a given graph $H$. A graph $G$ is said to be an $(a, d)$-$H$-antimagic total covering if there exist a bijective function $f: V(G) \cup E(G) \rightarrow \{1, 2,\dots ,|V (G)| + |E(G)|\}$ such that for all subgraphs $H'$ isomorphic to $H$, the total $H$-weights $w(H)= \sum_{v\in V(H')}f(v)+\sum_{e\in E(H')}f(v)$ form an arithmetic sequence $\{a, a + d, a +2d,...,a+(s - 1)d\}$, where $a$ and $d$ are positive integers and $s$ is the number of all subgraphs $H'$ isomorphic to $H$. Such a covering is called super if $f: V(G) \rightarrow \{1, 2,\dots ,|V (G)|\}$. This paper will study the existence of super $(a, d)-H$- antimagic total covering of connected amalgamation of fan graph for feasible $d$.