Kasteleynian Adjacency-Matrix-Based Methods

The most general Kekule-structure-count method of the present type was devised by Kasteleyn [146], though there is slightly earlier work for different special cases [33,147]. This too involves certain matrices, most simply the graph adjacency matrices /4(G) with rows columns that are labelled by the sites of G and elements that are all 0 except those Aab=+ with a b adjacent sites in G. Then Kastelyn shows how for planar graphs to set up a signed version (G) of this matrix with half of its +1 elements replaced by -1 such that 

That is, if one proceeds around a ring of sites f(l),f(2)./( ) then 

Kasteleyn [146] describes how this odd orientation is readily achievable for any planar graph. For instance, if one inserts arrows on edges of G so that an arrow from a to b indicates SaA=+l while Sba= 1, then an example of one such odd orientation is 

For the special case of polyhex benzenoid structures (such as is of importance for chemical applications) the determinantal formula for K(G) holds but in fact one need not even go through the signing procedure, using just /4(G) in place of S (G), as was earlier noted by Dewar Longuet-Higgins [33]. 

---