Abstract
The general Gutman index was introduced by Das and Vetrı́k very recently. For a graph \(G\), the general Gutman index is defined by
\[
Gut_{\alpha, \beta}(G) = \sum_{\{u, v\} \subseteq V(G)} [d_G(u) d_G(v)]^\alpha [D_G(u, v)]^\beta\,,
\]
where \(\alpha, \beta \in \mathbb{R}\), \(D_G(u, v)\) denotes the distance between \(u\) and \(v\) in \(G\), and \(d_G(u)\) and \(d_G(v)\) denote the degrees of \(u\) and \(v\) in \(G\), respectively. We show that for some \(\alpha\) and \(\beta\), the general Gutman index decrease or increase with changing the adjacency of vertices. For some \(\alpha\) and \(\beta\), we characterize trees of given order with the largest or the smallest general Gutman index.
