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<p>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 <i>sum index</i> 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.</p>

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New Developments on Graph Sum Index

  • 期刊论文

  • 发表日期：

    2026年3月9日

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  • Dheer Noal Desai,
  • Runze Wang

摘要

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 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.