Abstract
This article gives bounds on a substantial number of BID (bond incident degree) indices for connected graphs in terms of their order, size, and maximum degree. The considered BID indices include, among others, the Sombor index (together with its reduced
version), atom-bond sum-connectivity index, symmetric division deg index, sum-connectivity index, harmonic index, and Randić index. All the graphs that attain the obtained bounds are also characterized. All the established bounds are valid
also for molecular graphs. A graph of order \( n \) and size \( m \) is called an \( (n,m) \)-graph. The obtained bounds provide a partial solution to the problem of finding graphs with extremum (considered) BID indices over the class
of all connected \( (n,m) \)-graphs with a fixed maximum degree under certain constraints.
