On the Extremal Graphs with Respect to Bond Incident Degree Indices

Many existing degree based topological indices can be classified as bond incident degree (BID) indices, whose general form is $BID(G)=\sum_{uv\in E(G)}$ $\Psi(d_{u},d_{v})$, where $uv$ is the edge connecting the vertices $u,v$ of the graph $G$, $E(G)$ is the edge set of $G$, $d_{u}$ is the degree of the vertex $u$ and $\Psi$ is a non-negative real valued (symmetric) function of $d_{u}$ and $d_{v}$. Here, it has been proven that if the extension of $\Psi$ to the interval $[0,\infty)$ satisfies certain conditions then the extremal $(n,m)$-graph with respect to the BID index (corresponding to $\Psi$) must contain at least one vertex of degree $n-1$. It has been shown that these conditions are satisfied for the general sum-connectivity index (whose special cases are: the first Zagreb index and the Hyper Zagreb index), for the general Platt index (whose special cases are: the first reformulated Zagreb index and the Platt index) and for the variable sum exdeg index. Applying aforementioned result, graphs with maximum values of the aforementioned BID indices among tree, unicyclic, bicyclic, tricyclic and tetracyclic graphs were characterized. Some of these results are new and the already existing results are proven in a shorter and more unified way.