Degree-based graphical indices of $ k $-cyclic graphs

Let $ G $ be a graph with edge set $ E(G) $. Let $ d_x $ denote the degree of a vertex $ x $ in $ G $. For a nonnegative integer $ k $, a connected graph of order $ n $ and size $ n+k-1 $ is called a $ k $-cyclic graph. This paper is concerned with $ k $-cyclic graphs and their graphical indices of the form $ BID_{f}(G) = \sum_{uv\in E(G)} f(d_u, d_v) $, where $ f $ is a symmetric function whose outputs are real numbers. Particularly, the graphs minimizing or maximizing $ BID_{f} $ among all $ k $-cyclic graphs with a given order are studied under certain constraints on $ f $. Various existing indices meet these constraints, and hence the obtained results hold for those indices; more precisely, one of the obtained results covers the recently developed elliptic Sombor and Zagreb-Sombor indices, while another result covers the recently introduced Euler-Sombor index.