Some bond incident degree indices of (molecular) graphs with fixed order and number of cut vertices
type: Item
instance of: scholarly article
title: Some bond incident degree indices of (molecular) graphs with fixed order and number of cut vertices
zbMATH Open document ID: 1486.05296
DOI: 10.1155/2021/9970254
author: Abeer M. Albalahi
author: Akbar Ali
author: Tayyba Zafar
author: Wael W. Mohammed
published in: Discrete Dynamics in Nature and Society
publication date: 2021-12-13T00:00:00Z
review text: Summary: A bond incident degree (BID) index of a graph \(G\) is defined as \(\sum_{u v \in E (G)} f (d_G(u), d_G (v))\), where \(d_G(w)\) denotes the degree of a vertex \(w\) of \(G\), \(E(G)\) is the edge set of \(G\), and \(f\) is a real-valued symmetric function. The choice \(f(d_G(u), d_G (v))= a^{d_G(u)}+ a^{d_G(v)}\) in the aforementioned formula gives the variable sum exdeg index \(\mathrm{SEI}_a\), where \(a\neq 1\) is any positive real number. A cut vertex of a graph \(G\) is a vertex whose removal results in a graph with more components than \(G\) has. A graph of maximum degree at most 4 is known as a molecular graph. Denote by \(\mathbb{V}_{n,k}\) the class of all \(n\)-vertex graphs with \(k\geq 1\) cut vertices and containing at least one cycle. Recently, \textit{J. Du} and \textit{X. Sun}, characterized the graphs having the maximum value of \(\mathrm{SEI}_a\) from the set \(\mathbb{V}_n^k\) for \(a>1\) [``On the graph connectivity and the variable sum exdeg index'', AIMS Math. 6, No. 1, 607--622 (2021; \url{doi:10.3934/math.2021037})]. In the present paper, we not only characterize the graphs with the minimum value of \(\mathrm{SEI}_a\) from the set \(\mathbb{V}_n^k\) for \(a>1\), but we also solve a more general problem concerning a special type of BID indices. As the obtained extremal graphs are molecular graphs, they remain extremal if one considers the class of all \(n\)-vertex molecular graphs with \(k\geq 1\) cut vertices and containing at least one cycle.
Mathematics Subject Classification ID: 05C92
Mathematics Subject Classification ID: 92E10
Mathematics Subject Classification ID: 05C35
Mathematics Subject Classification ID: 05C07
Mathematics Subject Classification ID: 05C09
zbMATH DE Number: 7443992
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2021/9970254
OpenAlex ID: W3217333095
cites work: Bond incident degree (BID) indices of polyomino chains: a unified approach
cites work: Extremal polygonal cacti for bond incident degree indices
cites work: Q5165186
cites work: Extremal graphs with respect to variable sum exdeg index via majorization
cites work: On the variable sum exdeg index and cut edges of graphs
cites work: A note on the variable sum exdeg index/coindex of trees
cites work: On variable sum exdeg indices of quasi-tree graphs and unicyclic graphs
cites work: On the graph connectivity and the variable sum exdeg index
cites work: Q3369105
cites work: On multiplicative Zagreb indices of graphs
cites work: Q5165210
cites work: Asymptotic Laplacian-energy-like invariant of lattices
cites work: Q5422499
cites work: Q3451017
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description: scientific article; zbMATH DE number 7443992