Some new results on sum index and difference index

Let $G=(V(G),E(G))$ be a graph with a vertex set $V(G)$ and an edge set $E(G)$. For every injective vertex labeling $f:V\left(G\right)\to \mathbb{Z}$, there are two induced edge labelings denoted by $f^{+}:E\left(G\right)\to \mathbb{Z}$ and $f^{-}:E\left(G\right)\to \mathbb{Z}$. These two edge labelings $f^{+}$ and $f^{-}$ are defined by $f^{+}(uv)=f(u)+f(v)$ and $f^{-}(uv)=\left|f(u)-f(v)\right|$ for each $uv\in E(G)$ with $u,v\in V(G)$. The sum index and difference index of $G$ are induced by the minimum ranges of $f^{+}$ and $f^{-}$, respectively. In this paper, we obtain the properties of sum and difference index labelings. We also improve the bounds on the sum indices and difference indices of regular graphs and induced subgraphs of graphs. Further, we determine the sum and difference indices of various families of graphs such as the necklace graphs and the complements of matchings, cycles and paths. Finally, we propose some conjectures and questions by comparison.