01.06.2015 | Original Research | Ausgabe 1-2/2015
Extremal values of vertex-degree-based topological indices over graphs
- Zeitschrift:
- Journal of Applied Mathematics and Computing > Ausgabe 1-2/2015
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.
