The Generalized Laplacian Matrix

Gutman et al. (1983) proposed a simple method for computing the number of spanning trees of planar polycyclic graphs. It is based on the vertex-weighted inner dual 

FIGURE 2.24 Anthracene graph Gjq and its vertex-weighted inner dual Gu. 

The generalized Laplacian matrix, that is, the Laplacian matrix of the weighted inner dual, is given by 

The application of this procedure is demonstrated below for the anthracene graph and its weighted inner dual. The A and A matrices of the weighted inner dual Gu are as follows  

FIGURE 2.25 All spanning trees for the graph representing the carbon skeleton of naphthalene. 

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