Graphical edge-weight-function indices of trees

Akbar Ali; Sneha Sekar; Selvaraj Balachandran; Suresh Elumalai; Abdulaziz M. Alanazi; Taher S. Hassan; Yilun Shang; Akbar Ali; Sneha Sekar; Selvaraj Balachandran; Suresh Elumalai; Abdulaziz M. Alanazi; Taher S. Hassan; Yilun Shang

doi:10.3934/math.20241559

AIMS Mathematics

2024, Volume 9, Issue 11: 32552-32570. doi: 10.3934/math.20241559

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Graphical edge-weight-function indices of trees

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il, Saudi Arabia
2.
Department of Mathematics, School of Arts Sciences Humanities & Education, SASTRA Deemed University, Thanjavur, India
3.
Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Chengalpet 603 203, India
4.
Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia
5.
Department of Computer and Information Sciences, Northumbria University, Newcastle, NE1 8ST, UK

Received: 31 August 2024 Revised: 10 October 2024 Accepted: 30 October 2024 Published: 18 November 2024
MSC : 05C05, 05C07, 05C09

Consider a tree graph $ G $ with edge set $ E(G) $. The notation $ d_G(x) $ represents the degree of vertex $ x $ in $ G $. Let $ \mathfrak{f} $ be a symmetric real-valued function defined on the Cartesian square of the set of all distinct elements of the degree sequence of $ G $. A graphical edge-weight-function index for the graph $ G $, denoted by $ \mathcal{I}_\mathfrak{f}(G) $, is defined as $ \mathcal{I}_\mathfrak{f}(G) = \sum_{st \in E(G)} \mathfrak{f}(d_G(s), d_G(t)) $. This paper establishes the best possible bounds for $ \mathcal{I}_\mathfrak{f}(G) $ in terms of the order of $ G $ and parameter $ \mathfrak{p} $, subject to specific conditions on $ \mathfrak{f} $. Here, $ \mathfrak{p} $ can be one of the following three graph parameters: (ⅰ) matching number, (ⅱ) the count of pendent vertices, and (ⅲ) maximum degree. We also characterize all tree graphs that achieve these bounds. The constraints considered for $ \mathfrak{f} $ are satisfied by several well-known indices. We specifically illustrate our findings by applying them to the recently introduced Euler-Sombor index.
- topological index,
- graphical edge-weight-function index,
- Euler-Sombor index,
- tree graph,
- bound
Citation: Akbar Ali, Sneha Sekar, Selvaraj Balachandran, Suresh Elumalai, Abdulaziz M. Alanazi, Taher S. Hassan, Yilun Shang. Graphical edge-weight-function indices of trees[J]. AIMS Mathematics, 2024, 9(11): 32552-32570. doi: 10.3934/math.20241559

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Abstract

Consider a tree graph $ G $ with edge set $ E(G) $. The notation $ d_G(x) $ represents the degree of vertex $ x $ in $ G $. Let $ \mathfrak{f} $ be a symmetric real-valued function defined on the Cartesian square of the set of all distinct elements of the degree sequence of $ G $. A graphical edge-weight-function index for the graph $ G $, denoted by $ \mathcal{I}_\mathfrak{f}(G) $, is defined as $ \mathcal{I}_\mathfrak{f}(G) = \sum_{st \in E(G)} \mathfrak{f}(d_G(s), d_G(t)) $. This paper establishes the best possible bounds for $ \mathcal{I}_\mathfrak{f}(G) $ in terms of the order of $ G $ and parameter $ \mathfrak{p} $, subject to specific conditions on $ \mathfrak{f} $. Here, $ \mathfrak{p} $ can be one of the following three graph parameters: (ⅰ) matching number, (ⅱ) the count of pendent vertices, and (ⅲ) maximum degree. We also characterize all tree graphs that achieve these bounds. The constraints considered for $ \mathfrak{f} $ are satisfied by several well-known indices. We specifically illustrate our findings by applying them to the recently introduced Euler-Sombor index.

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