Sum index and difference index of graphs

TL;DR

This paper investigates the sum and difference indices of graphs, establishing bounds, exact values for specific graph families, and proposing a conjecture linking these two indices.

Contribution

It introduces bounds for the sum and difference indices, computes these indices for various graph families, and proposes a conjecture relating them.

Findings

Established bounds for sum and difference indices.

Determined indices for specific graph families.

Proposed a conjecture linking the two indices.

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.