Laplacian matrix

Before we start to calculate the Laplacian matrix we define the diagonal matrix DEG of a graph G. The non-diagonal elements are equal to zero. The matrix element in row i and column i is equal to the degree of vertex v/. 

As both the diagonal matrix DEG and the adjacency matrix A are symmetric it follows that the Laplacian matrix (Eq. (12)) is also a symmetric one. 

Other topological matrices are derived from the adjacency matrix, such as -> atom connectivity matrices, -> Laplacian matrix and the powers of the adjacency matrix used to obtain walk counts and the corresponding - molecular descriptors. 

Kirchhoff matrix Laplacian matrix Kirchhoff number -> resistance matrix Klopman-Henderson cumulative substructure count... 

Laplacian matrix (L) ( admittance matrix, Kirchhoff matrix)... 

The diagonalization of the Laplacian matrix gives A real eigenvalues "ki which constitute the Laplacian spectrum [Mohar, 1991b Trinajstic et al, 1994] and are conventionally labelled so that... 

The product of the positive A-l eigenvalues of the Laplacian matrix gives the spanning tree number T of the molecular graph Q as ... 

Also derived from the Laplacian matrix are the Mohar indices (TI)j and (TI)2, defined as ... 

Laplacian spectrum - Laplacian matrix lateral validation -> validation techniques... 

The resistance distance between any pair of vertices can also be calculated by the Laplacian matrix as the following ... 

The Kirchhoff number can also be directly calculated from the Laplacian matrix by the following ... 

Klein et al.216 have also shown that the combinatorial Laplacian matrix L (often called just the Laplacian matrix ) is related to matrix // ... 

The difference matrix I-H is called the normalized Laplacian matrix Lnorm (also sometimes called just the Laplacian matrix) of G and there is much theory about it.230 The matrix Lnorm is clearly also related to the connectivity index ... 

An interesting relationship between the vertex-edge incidence matrix, the edge-vertex incidence matrix and the Laplacian matrix L was found as... 

Laplacian matrix (L) (= admittance matrix, Kirchhoffmatrix, combinatorial Laplacian matrix) This is a square Ax A symmetric matrix, A being the number of vertices in the molecular graph, obtained as the difference between the vertex degree matrix V and the adjacency matrix A [Mohar, 1989b, 1989a] ... 

The Laplacian matrix is also related to the vertex-edge and edge-vertex incidence matrices. 

With some analogy to the Laplacian matrix is the second path matrix, denoted by S and defined as [John and Diudea, 2004]... 

Laplacian polynomial —> characteristic polynomial-based descriptors > Laplacian spectrum —> Laplacian matrix... 

Laplacian matrix defined as the difference adjacency matrix A other difference matrices + Wiener difference matrix, —> Szeged difference... 

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