@article{Ali2025, 
author = {Akbar Ali and Fehaid Salem Alshammari and Abdulaziz M. Alanazi and Taher S. Hassan},
title = {On degree-based graph invariants of fixed-order unicyclic graphs with prescribed maximum degree},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {10},
pages = {24500-24513},
keywords = {bond incident degree indices, unicyclic graph, extremal problem, maximum degree},
url = {https://www.sciopen.com/article/10.3934/math.20251086},
doi = {10.3934/math.20251086},
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    B  I      D          ϕ        (  G  )  =      ∑          y      z      ∈      E        ϕ  (      d    y    ,      d    z    ), where    ϕ 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    B  I      D    ϕ   among fixed-order unicyclic graphs with prescribed maximum degree, under specific assumptions on the function    ϕ. 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.}
}