Advertisement
Published in:
01-06-2015 | Original Research
Extremal values of vertex-degree-based topological indices over graphs
Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2015
Log inActivate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
Abstract
Given a graph \(G\) with \(n\) vertices, a vertex-degree-based topological index is defined from a set of real numbers \(\left\{ \varphi _{ij}\right\} \) as \( TI\left( G\right) =\sum m_{ij}\left( G\right) \varphi _{ij}\), where \( m_{ij}\left( G\right) \) is the number of edges between vertices of degree \(i\) and degree \(j\), and the sum runs over all \(1\le i\le j\le n-1\). In this paper we show that under certain conditions on the associated function \( \widehat{f}\left( i,j\right) =\frac{ij\varphi _{ij}}{i+j}\) and its partial derivatives, the extremal values of \(\widehat{f}\) are attained in the three points \(\left( 1,1\right) ,\left( 1,n-1\right) \) and \(\left( n-1,n-1\right) \) of the region \(\widehat{K}=\left\{ \left( i,j\right) \in {\mathbb {R}}\times {\mathbb {R}} :1\le i\le j\le n-1\right\} \), each of these points corresponding to \(K_{2}\cup K_{2}\cup \cdots \cup K_{2}\) (or \(K_{2}\cup K_{2}\cup \cdots \cup K_{2}\cup P_{3}\) if \(n\) is odd), the star \(S_{n}\) and the complete graph \(K_{n}\), respectively. As an application of this result, we find the extremal values of the well-known vertex-degree-based topological indices over \( {\mathcal {G}}_{n}\), the set of graphs with \(n\) non-isolated vertices.