On Unicyclic Graphs with a Given Number of Pendent Vertices or Matching Number and Their Graphical Edge-Weight-Function Indices

Akbar Ali (), Abdulaziz M. Alanazi, Taher S. Hassan and Yilun Shang
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Akbar Ali: Department of Mathematics, College of Science, University of Ha’il, Ha’il P.O. Box 2440, Saudi Arabia
Abdulaziz M. Alanazi: Department of Mathematics, Faculty of Sciences, University of Tabuk, Tabuk P.O. Box 741, Saudi Arabia
Taher S. Hassan: Department of Mathematics, College of Science, University of Ha’il, Ha’il P.O. Box 2440, Saudi Arabia
Yilun Shang: Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK

Mathematics, 2024, vol. 12, issue 23, 1-12

Abstract: Consider a unicyclic graph G with edge set E ( G ) . Let f be a real-valued symmetric function defined on the Cartesian square of the set of all distinct elements of G ’s degree sequence. A graphical edge-weight-function index of G is defined as I f ( G ) = ∑ x y ∈ E ( G ) f ( d G ( x ) , d G ( y ) ) , where d G ( x ) denotes the degree a vertex x in G . This paper determines optimal bounds for I f ( G ) in terms of the order of G and a parameter z , where z is either the number of pendent vertices of G or the matching number of G . The paper also fully characterizes all unicyclic graphs that achieve these bounds. The function f must satisfy specific requirements, which are met by several popular indices, including the Sombor index (and its reduced version), arithmetic–geometric index, sigma index, and symmetric division degree index. Consequently, the general results obtained provide bounds for several well-known indices.

Keywords: topological index; bond incident degree index; graphical edge-weight-function index; degree-based index; unicyclic graph; matching number; pendent vertex; bound (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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