Bond incident degree indices of fixed-order unicyclic graphs

A connected graph with the same order and size is known as a unicyclic graph. For a vertex u in a graph, we denote its degree by d_u. For a unicyclic graph G of a given order, we investigate an extremal problem concerning the degree-based graph invariants of the form BID_f (G) =^∑_vw∈E(G) f(dv, dw), where E(G) represents the edge set of G and f is a symmetric non-negative function that depends on the degrees of adjacent vertices of G. These graph invariants are known as bond incident degree indices. One of our results provides a partial solution to an open problem proposed by Ergotić and Došlić in [MATCH Commun. Math. Comput. Chem. 95 (2026) 265–283].