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 kcyclic graph. This paper is concerned with k-cyclic graphs and their graphical indices of the form BID_f (G) =^∑uv∈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.