Note on the Minimum Bond Incident Degree Indices of k-Cyclic Graphs

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) = Sigma(uv is an element of E(G)) I(d(u), d(v)), where I(x, y) > 0 is a symmetric real function with x >= 1 and y >= 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 >= 3, n >= 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].