On the general Z-type index of connected graphs

Let $G=(V,E)$ be a connected graph, and $d\left(u\right)$ the degree of vertex $u\in V$. We define the general $Z$-type index of $G$ as $Z_{\alpha ,\beta }\left(G\right)={\sum }_{uv\in E}{[d\left(u\right)+d\left(v\right)-\beta ]}^{\alpha }$, where $\alpha$ and $\beta$ are two real numbers. This generalizes several famous topological indices, such as the first and second Zagreb indices, the general sum-connectivity index, the reformulated first Zagreb index, and the general Platt index, which have successful applications in QSPR/QSAR research. Hence, we are able to study these indices in a unified approach.

Let $C\left(\pi \right)$ the set of connected graphs with degree sequence $\pi$. In the present paper, under different conditions of $\alpha$ and $\beta$, we show that:

$\left(1\right)$ There exists a so-called BFS-graph having extremal $Z_{\alpha ,\beta }$ index in $C\left(\pi \right)$;

$\left(2\right)$ If $\pi$ is the degree sequence of a tree, a unicyclic graph, or a bicyclic graph, with minimum degree 1, then there exists a special BFS-graph with extremal $Z_{\alpha ,\beta }$ index in $C\left(\pi \right)$;

$\left(3\right)$ The so-called majorization theorem of $Z_{\alpha ,\beta }$ holds for trees, unicyclic graphs, and bicyclic graphs.

As applications of the above results, we determine the extremal graphs with maximum $Z_{\alpha ,\beta }$ index for $\alpha >1$ and $\beta \le 2$ in the set of trees, unicyclic graphs, and bicyclic graphs with given number of pendent vertices, maximum degree, independence number, matching number, and domination number, respectively. These extend the main results of some published papers.