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)=∑uv∈E(G)f(du,dv), where uv is the edge connecting vertices u,v of the graph G, E(G) is the edge set of G, du is the degree of a vertex u and f is a non-negative real valued (symmetric) function of du and dv. Firstly, here an intuitively expected result is proven, which states that an extremal (n,m)-graph with respect to the BID index (corresponding to f) must contain at least one vertex of degree n−1 if f satisfies certain conditions. It is shown that these certain 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. With help of the aforementioned result of existence of at least one vertex of degree n−1 and further analysis, graphs with maximum values of the above mentioned BID indices among tree, unicyclic, bicyclic, tricyclic and tetracyclic graphs are characterized. Some of these results are new and the already existing results are proven in a shorter and more unified way.