neighborhood first zagreb index and maximal unicyclic and bicyclic graphs

The neighborhood first zagreb index $nm_{1}$ measures the topological properties of a molecular graph. neighborhood first zagreb index $nm_{1}$ is defined as $nm_{1}(g) = sum_{ vin v (g)}(s(v))^{2}$, where $s(v)$ used to represent the sum of degrees of vertices adjacent to a vertex $v$ in a graph $g$.  in this study, we focus on characterizing the graphs with the maximum neighborhood first zagreb index in the class of unicyclic/bicyclic graphs on $n$ vertices, where $n$ is a fixed integer greater than or equal to $5$. specifically, we are interested in identifying the graphs that have the highest value according to the recently introduced neighborhood first zagreb index $nm_{1}$.