Sum and integral sum graphs -- A survey

TL;DR

This survey reviews the properties and developments related to sum and integral sum graphs, which are defined by labeling vertices with integers such that edges correspond to sums within the label set.

Contribution

It compiles and discusses various properties and results of sum and integral sum graphs from multiple authors, providing a comprehensive overview.

Findings

Summarizes key properties of sum and integral sum graphs.

Highlights open problems and research directions.

Provides a unified view of existing results.

Abstract

Frank Harary introduced the concepts of sum and integral sum graphs. A graph $G$ is a \textit{sum graph} if the vertices of $G$ can be labeled with distinct positive integers so that $e = uv$ is an edge of $G$ if and only if the sum of the labels on vertices $u$ and $v$ is also a label in $G.$ An \textit{integral sum graph} is also defined just as sum graph, the difference being that the labels may be any distinct integers. In this survey article, the authors bring out several properties of sum and integral sum graphs obtained by different authors.