Super (a,d)-H-Antimagic Total Covering of Connected Semi Jahangir Graph

Abstract

Let $G$ be a finite, simple and undirected graph. A graph $G$ is called 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 $\{a, a + d, a +2d,...,a+(t - 1)d\}$, where $a$ and $d$ are positive integers and $t$ is the number of all subgraphs $H'$ isomorphic to $H$. Such a labeling is called super if $f: V(G) \rightarrow \{1, 2,\dots ,|V (G)|\}$. In this paper we study a super $(a, d)$-$C_4$-antimagic total covering of connected Semi Jahangir graph denoted by $SJ_n$.