The Spum and Sum-diameter of Graphs: Labelings of Sum Graphs

TL;DR

This paper explores the concepts of spum and sum-diameter in sum graphs, providing bounds, analyzing their behavior under various operations, and extending the ideas to hypergraphs.

Contribution

It introduces the sum-diameter as a natural alternative to spum, offers asymptotic bounds, and generalizes the concept to hypergraphs.

Findings

Derived asymptotically tight bounds for sum-diameter

Analyzed behavior under graph operations

Extended concepts to hypergraphs

Abstract

A sum graph is a finite simple graph whose vertex set is labeled with distinct positive integers such that two vertices are adjacent if and only if the sum of their labels is itself another label. The spum of a graph $G$ is the minimum difference between the largest and smallest labels in a sum graph consisting of $G$ and the minimum number of additional isolated vertices necessary so that a sum graph labeling exists. We investigate the spum of various families of graphs, namely cycles, paths, and matchings. We introduce the sum-diameter, a modification of the definition of spum that omits the requirement that the number of additional isolated vertices in the sum graph is minimal, which we believe is a more natural quantity to study. We then provide asymptotically tight general bounds on both sides for the sum-diameter, and study its behavior under numerous binary graph operations as well as vertex and edge operations. Finally, we generalize the sum-diameter to hypergraphs.