On the neighbourhood degree sum-based Laplacian energy of graphs

A useful extension of the Laplacian matrix is proposed here and the corresponding modification of the Laplacian energy ($LE$) is presented. The neighbourhood degree sum-based Laplacian energy ($L_{N}E$) is produced by means of the eigenvalues of the newly introduced neighbourhood degree sum-based Laplacian matrix ($L_N$). We investigate the mathematical properties of $L_{N}E$ by comparing it with the Laplacian energy. The role of $L_{N}E$ in structure–property modelling of molecules is demonstrated by statistical approach. The formulation of $L_{N}E$ is not ad hoc; rather, its chemical significance exerts that it outperforms $LE$ in modelling physiochemical behaviours of molecules. Mathematical properties of $L_N$ are also revealed by finding crucial bounds of its eigenvalues with characterizing extremal graphs.