Sum index of Cartesian product networks

Shuo Liu; Zhao Wang; Shuo Liu; Zhao Wang

doi:10.3934/math.20261186

AIMS Mathematics

2026, Volume 11, Issue 9: 29888-29906. doi: 10.3934/math.20261186

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Research article

Sum index of Cartesian product networks

Shuo Liu ¹,
Zhao Wang ^{2
,
,}

1.
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom
2.
College of Science, China Jiliang University, Hangzhou 310018, China

Received: 20 July 2026 Revised: 08 September 2026 Accepted: 10 September 2026 Published: 15 September 2026
MSC : 05C40, 05C78

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 sum index of a graph is defined as the minimum number of distinct such sums over all injective vertex labelings. In this paper, we establish upper bounds for the sum index of Cartesian product graphs. As applications of our results, we determine the exact values of the sum index of grid networks.
- sum index,
- graph labeling,
- Cartesian product graphs,
- networks
Citation: Shuo Liu, Zhao Wang. Sum index of Cartesian product networks[J]. AIMS Mathematics, 2026, 11(9): 29888-29906. doi: 10.3934/math.20261186

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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 sum index of a graph is defined as the minimum number of distinct such sums over all injective vertex labelings. In this paper, we establish upper bounds for the sum index of Cartesian product graphs. As applications of our results, we determine the exact values of the sum index of grid networks.

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