Abstract
Let \( G\) be a connected graph with \( n\) vertices. The bond incident degree (BID) indices \( TI(G)\) of \( G\) with edge-weight function \( I(x,y)\) is defined as
\[
TI(G)=\sum\limits_{uv\in E(G)}I(d_{u},d_{v}),
\]
where \( I(x,y)>0 \) is a symmetric real function with \( x\geq 1\) and \( y\geq 1\), \( d_{u}\) is the degree of vertex \( u\) in \( G\).
In this note, we deduce a number of previously established results, and state a few new. For the BID index \( TI\) with the property \( P^{*}\), we can obtain the minimum \( k\)-cyclic (chemical) graphs for \( k\geq3\), \( n\geq 5(k-1)\). These BID indices include the Sombor index, the general Sombor index, the \( p\)-Sombor index, the general sum-connectivity index and so on. Thus this note extends the results of Liu et al. [H. Liu, L. You, Y. Huang, Sombor index of c-cyclic chemical graphs, MATCH Commun. Math. Comput. Chem. 90 (2023) 495-504] and Ali et al. [A. Ali, D. Dimitrov, Z. Du, F. Ishfaq, On the extremal graphs for general sum-connectivity index \( (\chi_{\alpha})\) with given cyclomatic number when \( \alpha>1\), Discrete Appl. Math. 257 (2019) 19-30].
