BCUT (aka Burden Eigenvalues)[2,3,14]

These descriptors are the eigenvalues of a modified connectivity matrix, the Burden matrix $\varmathbb{B}$. The $\varmathbb{B}$ matrix is an H depleted molecular graph defined as

• diagonal elements are atomic numbers, $Z_i$, of the elements
• off diagonal elements, $B_{ij}$, representing bonded atoms $i$ and $j$ are equal to $\pi^* \cdot 10^{-1}$ where $\pi^*$ is the conventional bond order (ie 1,2,3,1.5 for single, double, triple and aromatic bonds respectively)
• Off diagonal elements corresponding to terminal bonds are augmented by .01
• All other matrix elements are set to .001

The ordered sequence of the $n$ smallest eigenvalues of $\varmathbb{B}$ was proposed as a molecular descriptor based on the assumption that the lowest eigenvalues contain contributions from all the atoms and thus reflect topology of the molecule.

The BCUT descriptors are an extension of the Burden eigenvalues and consider 3 classes of matrices whose diagonal elements correspond to

1. atomic charge related values
2. atomic polarizability related values
3. atomic H bond abilities

A variety of definition have been used for the off diagonal terms and both 2D and 3D approaches are considered. The highest and lowest eigenvalues of these matrices have been shown to be discriminating descriptors.