Authors:
Juan Rada, José M. Rodríguez, José M. Sigarreta
Abstract
Let \( G \) be a graph with vertex set \( V \) and edge set \( E \). A topological index has the form
\begin{equation*}
TI\left(G \right) = \sum_{uv \in E}f\left(d_{u},d_{v} \right),
\end{equation*}
where \( f=f\left(x,y \right) \) is a pertinently chosen function which must be
symmetric and real-valued for all \( x, y \) pertaining to vertex degrees of the graph \( G \). Particularly interesting are the
Sombor index \( \mathcal{SO} \) and the elliptic Sombor index \( \mathcal{ESO} \), induced by the functions \( f\left(x,y \right)
=\sqrt{x^{2}+y^{2}} \) and \( f\left(x,y \right) = \left(x+y \right)\sqrt{x^{2}+y^{2}} \), respectively.
In this paper we solve some optimization problems for the general elliptic Sombor index \( \mathcal{ESO}_{a} \),
induced by the function \( f(x,y) = (x+y)^{a}(x^2+y^2)^{a/2} \) \( (a \neq 0) \),
in particular on the set of graphs (respectively, trees) with \( n \) vertices.
