Abstract
A bond incident degree (BID) index of a graph \(G\) is defined
as \(f(d_G(u), d_G(v))\), with summation ranging over all pairs of
adjacent vertices \(u, v\) of \(G\), where \(d_G(w)\) denotes the
degree of the vertex \(w\) of \(G\), and \(f\) is a real-valued
symmetric function. This paper reports extremal results for BID
indices of the type \(I_{f_i}(G) = \sum [f_i(d_G(u))/d_G(u) +
f_i(d_G(v))/d_G(v)]\), where \(i \in {1, 2}\), \(f_1\) is strictly
convex, and \(f_2\) is strictly concave. Graphs attaining minimum
\(I_{f_1}\) and maximum \(I_{f_2}\) are characterized from the class
of connected \((n, m)\)-graphs and chemical \((n, m)\)-graphs, where
\(n\) and \(m\) satisfy the conditions \(3n \geq 2m\), \(n \geq 4\),
\(m \geq n + 1\). By this, we extend and complement the recent result
by Tomescu [ MATCH Commun. Math. Comput. Chem.85 (2021)
285-294], and cover several well-known indices, including general
zeroth-order RandiÄ‡ index, multiplicative first and second Zagreb
indices, variable sum exdeg index, and Lanzhou index.