Abstract
For a graph \(G\), we generalize the well-known Gutman index by introducing the general Gutman index
\[
Gut_{a,b} (G) = \sum_{ \{ u , v \} \subseteq V(G)} [d_{G} (u) d_{G} (v)]^a [D_{G} (u, v) ]^{b}\,,
\]
where \(a,b \in \mathbb{R}\), \(D_{G} (u, v)\) is the distance between vertices \(u\) and \(v\) in \(G\), and \(d_{G} (u)\) and \(d_{G} (v)\) are the degrees of \(u\) and \(v\), respectively. We show that for some \(a\) and \(b\), the \(Gut_{a,b}\) index decreases/increases with the addition of edges. We present sharp bounds on the general Gutman index for multipartite graphs of given order, graphs of given order and chromatic number, and starlike trees of given order and maximum degree. We also state several problems open for further research.
