Skew-symmetric adjacency matrix

The skew-symmetric adjacency matrix S of a graph G is defined with respect to oriented edges (indicated by arrows) the edge e betwee opposite sense v - u. Then S is defined by ... 

Chart 2.3. The skew-symmetric adjacency matrix of [12]annulene and its partitioning notice the indicated coloring, numbering of vertices, and orientation of edges (in terms of arrows). 

The method under consideration, which is based on the Pfaffian of a skew-symmetric adjacency matrix, has been employed in a computer program (Cyvin SJ, BrunvoU and Cyvin 1990c). Thus it became possible to execute routine computations of Kekule number counts for single coronoids of type (b). 

Method of the Pfaffian to a Skew-Symmetric Matrix, Consider [12]annulene (see Chart 2), which is a [4k]—cycle, in contrast to [lOjannulene (Chart 1) being a [4k + 2]-cyde. The Kekule structure count is again K = 2, but the adjacency matrix (as well as its submatrices according to the partitioning shown in Chart 2) appear to be singular. Hence eqn. (13) would give zero. A correction is needed. 

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