Optimum Sum Labeling of Finite Union of Sum Graphs

Abstract

Let $G=(V,E)$ be a simple, finite, and undirected graph. A sum labeling is a one-to-one mapping $L$ from a set of vertices of $G$ to a finite set of positive integers $S$ such that if $u$ and $v$ are vertices of $G$, then $uv$ is an edge in $G$ if and only if there is a vertex $w$ in $G$ and $L(w)=L(u)+L(v)$. A graph $G$ that has a sum labeling is called a sum graph. The minimal isolated vertex that is needed to make $G$ a sum labeling is called the sum number of $G$, denoted as $\sigma (G)$. The sum number of a sum graph $G$ is always greater than or equal to $\delta (G)$, the minimum degree of $G$. An optimum sum graph is a sum graph that has $\sigma (G)=\delta (G)$. In this paper, we discuss sum numbers of finite unions of some families of optimum sum graphs, such as cycles and friendship graphs.