On degree-based graph invariants of fixed-order unicyclic graphs with prescribed maximum degree

Akbar Ali; Fehaid Salem Alshammari; Abdulaziz M. Alanazi; Taher S. Hassan; Akbar Ali; Fehaid Salem Alshammari; Abdulaziz M. Alanazi; Taher S. Hassan

doi:10.3934/math.20251086

AIMS Mathematics

2025, Volume 10, Issue 10: 24500-24513. doi: 10.3934/math.20251086

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On degree-based graph invariants of fixed-order unicyclic graphs with prescribed maximum degree

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il, Saudi Arabia
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
3.
Department of Mathematics, Faculty of Sciences, University of Tabuk, Tabuk, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 25 June 2025 Revised: 20 September 2025 Accepted: 13 October 2025 Published: 27 October 2025
MSC : 05C07, 05C09, 05C35

Consider a graph $ G $ having edge set $ E $, and denote by $ d_x $ the degree of a vertex $ x $ in $ G $. A unicyclic graph is defined as a connected graph containing exactly one cycle. This work focuses on unicyclic graphs of a fixed order and examines the graph invariants of such graphs of the form $ BID_{\phi}(G) = \sum_{yz \in E} \phi(d_y, d_z) $, where $ \phi $ is a symmetric and real-valued function. Such graph invariants are known as BID (bond incident degree) indices. The main objective is to determine the graphs that either minimize or maximize the quantity $ BID_\phi $ among fixed-order unicyclic graphs with prescribed maximum degree, under specific assumptions on the function $ \phi $. These assumptions are satisfied by several classical and modern degree-based graph invariants. Generally, the results obtained are applicable to a wide range of such invariants. In particular, one of the obtained results covers the harmonic and sum-connectivity indices, while another applies to the recently proposed Sombor and Euler–Sombor indices as well as their reduced versions.
- bond incident degree indices,
- unicyclic graph,
- extremal problem,
- maximum degree
Citation: Akbar Ali, Fehaid Salem Alshammari, Abdulaziz M. Alanazi, Taher S. Hassan. On degree-based graph invariants of fixed-order unicyclic graphs with prescribed maximum degree[J]. AIMS Mathematics, 2025, 10(10): 24500-24513. doi: 10.3934/math.20251086

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Abstract

Consider a graph $ G $ having edge set $ E $, and denote by $ d_x $ the degree of a vertex $ x $ in $ G $. A unicyclic graph is defined as a connected graph containing exactly one cycle. This work focuses on unicyclic graphs of a fixed order and examines the graph invariants of such graphs of the form $ BID_{\phi}(G) = \sum_{yz \in E} \phi(d_y, d_z) $, where $ \phi $ is a symmetric and real-valued function. Such graph invariants are known as BID (bond incident degree) indices. The main objective is to determine the graphs that either minimize or maximize the quantity $ BID_\phi $ among fixed-order unicyclic graphs with prescribed maximum degree, under specific assumptions on the function $ \phi $. These assumptions are satisfied by several classical and modern degree-based graph invariants. Generally, the results obtained are applicable to a wide range of such invariants. In particular, one of the obtained results covers the harmonic and sum-connectivity indices, while another applies to the recently proposed Sombor and Euler–Sombor indices as well as their reduced versions.

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