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 \(d_{u_i}+d_{u_j}\), if \(u_i\) is connected to \(u_j\); 0, otherwise, where \(d_{u_i}\) is degree of i-th vertex. The current work is concerned with the mathematical properties and chemical significance of the spectral radius (\(\rho _1\)) associated with this matrix. The lower and upper bounds of \(\rho _1\) 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 \(\rho _1\) is also explained.