Super $(a,d)$-Edge-Antimagic Total Labelings of Generalized Friendship Graphs

Abstract

An $(a,d)$-edge-antimagic total labeling of a graph $G$ with $p$ vertices and $q$ edges is a bijection $f$ from the set of all vertices and edges to the set of positive integers $\{1,2,3,\dots ,p+q\}$ such that all the edge-weights $w(uv)=f(u)+f(v)+f(uv)$ for $uv\in E(G)$, form an arithmetic progression starting from $a$ and having common difference $d$. An $(a,d)$-edge-antimagic total labeling is called a super $(a,d)$-edge-antimagic total labeling ($(a,d)$-SEAMT labeling) if $f(V(G))=\{1,2,3,\dots ,p\}$. The graph $F_n$, consisting of $n$ triangles with a common vertex, is called the friendship graph. The generalized friendship graph $F_{m_1,m_2,\dots ,m_n}$ consists of $n$ cycles of orders $m_1\le m_2\le \cdots \le m_n$ having a common vertex. In this paper, we prove that the friendship graph $F_{16}$ does not admit a $(a,2)$-SEAMT labeling. We also investigate the existence of $(a,d)$-SEAMT labeling for several classes of generalized friendship graphs.