Degree-based function index of graphs with given bipartition and small cyclomatic number

We investigate the degree-based function index If (G) = P vw∈E(G) f(dG(v), dG(w)) of a graph G, where E(G) is the set of edges of G, dG(v) and dG(w) are the degrees of vertices v and w in G, respectively, and f is a symmetric function of two variables which satisfies some conditions. We obtain sharp upper bounds on If for trees, unicyclic graphs and bicyclic graphs with given bipartition. Then, among trees and unicyclic graphs with given bipartition, we present graphs with the largest values of the first general Gourava index F GOa(G) = P vw∈E(G) [dG(v)dG(w) + dG(v) + dG(w)]a for a ≥ 1, Bollobas- ´ Erdos-Sarkar index ˝ BESl,a(G) = P vw∈E(G) [(dG(v) + l)(dG(w) + l)]a for a ≥ 1 and l > −1 (with its special cases which are general reduced second Zagreb index GRMl(G) = P vw∈E(G) (dG(v) + l)(dG(w) + l) for l > −1, and general Randic´ index Ra(G) = P vw∈E(G) [dG(v)dG(w)]a for a ≥ 1), general Sombor index SOa,b(G) = P vw∈E(G) ([dG(v)]a + [dG(w)]a ) b , generalized Zagreb index GZa,b(G) = P vw∈E(G) ([dG(v)]a [dG(w)]b+[dG(v)]b [dG(w)]a ) and one other general index Ma,b(G) = P vw∈E(G) [dG(v)dG(w)]a [dG(v) + dG(w)]b for a ≥ 1 and b ≥ 1.