Sum index and difference index of graphs

Abstract

Let $G$ be a nonempty simple graph with a vertex set $V(G)$ and an edge set $E(G)$. For every injective vertex labeling $f:V(G)\to \mathbb{Z}$, there are two induced edge labelings, namely $f^{+}:E(G)\to \mathbb{Z}$ defined by $f^{+}(uv)=f(u)+f(v)$, and $f^{-}:E(G)\to \mathbb{Z}$ defined by $f^{-}(uv)=|f(u)-f(v)|$. The sum index and the difference index are the minimum cardinalities of the ranges of $f^{+}$ and $f^{-}$, respectively. We provide upper and lower bounds on the sum index and difference index, and determine the sum index and difference index of various families of graphs. We also provide an interesting conjecture relating the sum index and the difference index of graphs.