New developments on graph sum index

TL;DR

This paper explores the graph sum index, linking it to additive combinatorics, and determines its values for various graph classes while studying related extremal edge problems.

Contribution

It introduces new results on the sum index for specific graphs and connects the concept to additive combinatorics and extremal graph theory.

Findings

Sum indices of complete multipartite graphs determined

Sum indices of hypercubes calculated

Maximum edges in graphs with fixed sum index studied

Abstract

In a graph, we assign distinct integers to the vertices, and take the sum of two integers if they are on two adjacent vertices. The minimum possible number of different sums is the \emph{sum index} of this graph. In this paper, we present some new developments on graph sum index. First, we explain the connections between graph sum index and results in additive combinatorics. Then, we determine the sum indices of the complete multipartite graphs, hypercubes, and some cluster graphs. Also, we study the maximum number of edges in a graph with a fixed sum index, which is related to the forbidden subgraph problem.