Neighborhood first Zagreb index and maximal unicyclic and bicyclic graphs

The Neighborhood First Zagreb Index NM1 measures the topological properties of a molecular graph. Neighborhood First Zagreb Index NM1 is defined as NM1(G) = Ev is an element of 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 NM1.