First zagreb spectral radius of unicyclic graphs and trees

In light of the successful investigation of the adjacency matrix, a significant amount of its modification is observed employing numerous topological indices. The matrix corresponding to the well-known first Zagreb index is one of them. The entries of the first Zagreb matrix are du_i+du_j, if u_i is connected to u_j; 0, otherwise, where du_i is degree of i-th vertex. The current work is concerned with the mathematical properties and chemical significance of the spectral radius (ρ₁) associated with this matrix. The lower and upper bounds of ρ₁ are computed with characterizing extremal graphs for the class of unicyclic graphs and trees. The chemical connection of the first Zagreb spectral radius is established by exploring its role as a structural descriptor of molecules. The isomer discrimination ability of ρ₁ is also explained.