Some Bond Incident Degree Indices of (Molecular) Graphs with Fixed Order and Number of Cut Vertices

A bond incident degree (BID) index of a graph G is defined as ∑uv∈EGfdGu,dGv, where dGw denotes the degree of a vertex w of G, EG is the edge set of G, and f is a real-valued symmetric function. The choice fdGu,dGv=adGu+adGv in the aforementioned formula gives the variable sum exdeg index SEIa, where a≠1 is any positive real number. A cut vertex of a graph G is a vertex whose removal results in a graph with more components than G has. A graph of maximum degree at most 4 is known as a molecular graph. Denote by Vn,k the class of all n-vertex graphs with k≥1 cut vertices and containing at least one cycle. Recently, Du and Sun [AIMS Mathematics, vol. 6, pp. 607–622, 2021] characterized the graphs having the maximum value of SEIa from the set Vnk for a>1. In the present paper, we not only characterize the graphs with the minimum value of SEIa from the set Vnk for a>1, but we also solve a more general problem concerning a special type of BID indices. As the obtained extremal graphs are molecular graphs, they remain extremal if one considers the class of all n-vertex molecular graphs with k≥1 cut vertices and containing at least one cycle.