Extremal c-cyclic graphs with respect to the general multiplicative first Zagreb index

The general multiplicative first Zagreb index of a simple graph H is expressed as the product of the weights (deg_H (ω))^α over all vertices ω of H, where deg_H (ω) shows the degree of ω, and α ≠ 0 is a real number. The cyclomatic number of a connected graph H is given by c = ϵ − ν + 1, where ϵ and ν are the size and order of H, respectively. In this paper, we present sharp bounds for the general multiplicative first Zagreb index of simple connected graphs with cyclomatic number c focusing on the cases when c=0, 1, and 2. We also extend our findings to molecular trees and to all simple connected graphs with the maximum degree ∆ and cyclomatic number c, where ∆ ≥ 2c. In addition, we identify the graphs reaching these bounds.