Degree-based graphical indices of $ k $-cyclic graphs

Akbar Ali; Darko Dimitrov; Tamás Réti; Abdulaziz Mutlaq Alotaibi; Abdulaziz M. Alanazi; Taher S. Hassan; Akbar Ali; Darko Dimitrov; Tamás Réti; Abdulaziz Mutlaq Alotaibi; Abdulaziz M. Alanazi; Taher S. Hassan

doi:10.3934/math.2025609

AIMS Mathematics

2025, Volume 10, Issue 6: 13540-13554. doi: 10.3934/math.2025609

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Degree-based graphical indices of $ k $-cyclic graphs

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il, Saudi Arabia
2.
Faculty of Information Studies, Novo Mesto, Slovenia
3.
Óbuda University, Bécsiút, Budapest, Hungary
4.
Department of Mathematics, College of Science and Humanities in AlKharj, Prince Sattam Bin Abdulaziz University, AlKharj 11942, Saudi Arabia
5.
Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia
6.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 24 March 2025 Revised: 26 May 2025 Accepted: 04 June 2025 Published: 12 June 2025
MSC : 05C07, 05C09

Let $ G $ be a graph with edge set $ E(G) $. Let $ d_x $ denote the degree of a vertex $ x $ in $ G $. For a nonnegative integer $ k $, a connected graph of order $ n $ and size $ n+k-1 $ is called a $ k $-cyclic graph. This paper is concerned with $ k $-cyclic graphs and their graphical indices of the form $ BID_{f}(G) = \sum_{uv\in E(G)} f(d_u, d_v) $, where $ f $ is a symmetric function whose outputs are real numbers. Particularly, the graphs minimizing or maximizing $ BID_{f} $ among all $ k $-cyclic graphs with a given order are studied under certain constraints on $ f $. Various existing indices meet these constraints, and hence the obtained results hold for those indices; more precisely, one of the obtained results covers the recently developed elliptic Sombor and Zagreb-Sombor indices, while another result covers the recently introduced Euler-Sombor index.
- graphical index,
- bond incident degree index,
- Euler-Sombor index,
- elliptic Sombor index,
- Zagreb-Sombor index,
- topological index,
- $ k $-cyclic graph
Citation: Akbar Ali, Darko Dimitrov, Tamás Réti, Abdulaziz Mutlaq Alotaibi, Abdulaziz M. Alanazi, Taher S. Hassan. Degree-based graphical indices of $ k $-cyclic graphs[J]. AIMS Mathematics, 2025, 10(6): 13540-13554. doi: 10.3934/math.2025609

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Abstract

Let $ G $ be a graph with edge set $ E(G) $. Let $ d_x $ denote the degree of a vertex $ x $ in $ G $. For a nonnegative integer $ k $, a connected graph of order $ n $ and size $ n+k-1 $ is called a $ k $-cyclic graph. This paper is concerned with $ k $-cyclic graphs and their graphical indices of the form $ BID_{f}(G) = \sum_{uv\in E(G)} f(d_u, d_v) $, where $ f $ is a symmetric function whose outputs are real numbers. Particularly, the graphs minimizing or maximizing $ BID_{f} $ among all $ k $-cyclic graphs with a given order are studied under certain constraints on $ f $. Various existing indices meet these constraints, and hence the obtained results hold for those indices; more precisely, one of the obtained results covers the recently developed elliptic Sombor and Zagreb-Sombor indices, while another result covers the recently introduced Euler-Sombor index.

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