Matching some graph dimensions with special generating functions

Ahmet Sinan Cevik; Ismail Naci Cangul; Yilun Shang; Ahmet Sinan Cevik; Ismail Naci Cangul; Yilun Shang

doi:10.3934/math.2025389

AIMS Mathematics

2025, Volume 10, Issue 4: 8446-8467. doi: 10.3934/math.2025389

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Matching some graph dimensions with special generating functions

1.
Department of Mathematics, Selcuk University, Konya 42130, Turkey
2.
Department of Mathematics, Bursa Uludag University, Bursa 16059, Turkey
3.
Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK

Received: 21 November 2024 Revised: 16 March 2025 Accepted: 07 April 2025 Published: 14 April 2025
MSC : 05C35, 05C80, 11B68, 12D10, 20F05, 26C10, 57M15

In this work, we investigated the relationship between special generating functions, such as array polynomials and graph dimensions, including metric, multiset, outer multiset, and local multiset dimensions, using minimal monoid presentations. The present paper is founded on earlier contributions and we are concerned with resolving the matching issue. In this context, our focus is on the matching between graph dimensions and generator functions, a subject that has not been examined and is alluded to in Open problem 3 in the paper by A. S. Cevik [Matching some graph dimensions with special presentations, Montes Taurus J. Pure Appl. Math., 6 (2024), 78-89]. As part of this effort, we address the characterization of graphs with infinite multiset dimensions and provide a partial classification based on outer multiset, local multiset, and metric dimensions.
- minimal presentation,
- graphs,
- metric dimension,
- multiset dimension,
- outer multiset dimension,
- local multiset dimension,
- array polynomials,
- sirling numbers
Citation: Ahmet Sinan Cevik, Ismail Naci Cangul, Yilun Shang. Matching some graph dimensions with special generating functions[J]. AIMS Mathematics, 2025, 10(4): 8446-8467. doi: 10.3934/math.2025389

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Abstract

In this work, we investigated the relationship between special generating functions, such as array polynomials and graph dimensions, including metric, multiset, outer multiset, and local multiset dimensions, using minimal monoid presentations. The present paper is founded on earlier contributions and we are concerned with resolving the matching issue. In this context, our focus is on the matching between graph dimensions and generator functions, a subject that has not been examined and is alluded to in Open problem 3 in the paper by A. S. Cevik [Matching some graph dimensions with special presentations, Montes Taurus J. Pure Appl. Math., 6 (2024), 78-89]. As part of this effort, we address the characterization of graphs with infinite multiset dimensions and provide a partial classification based on outer multiset, local multiset, and metric dimensions.

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