ABSTRACT
Dewar and Longnet-Higgins [57] have found an important relation ship between Hûckel theory and resonance theory: The determinant of A can be evaluated from a knowledge of the number of Kekulé structures of a conjugated molecule whose graph has the adjacency matrix A. Kekulé structures are principal resonance structures in the qualitative valence bond theory. The concept of Kekulé structures coincides with the concept of 1-factors in the mathematical literature [17]. The number of Kekulé structures is denoted by K. Dewar and Longuet-Higgins found that in the case of alternant hydrocarbons, Kekulé structures may be separated into two classes of different parity: even, ÎT+, and odd, K ~ , respectively. The same (even) or different (odd) parity is determined in the DewarLongnet-Higgins scheme according to whether the number of transposi tions of double bonds required to transform one of the structures into the other is even or odd. There is some difficulty in determining the parity of Kekulé structures by use of the Dewax-Longuet-Higgins method [60]. Accordingly, we have endeavoured to establish simple rules for determin ing the mutual parity of two Kekulé structures [16,58,60,61]. For an arbitrary conjugated system (subject to the limitation that it should not contain only odd-membered rings if it is a three-or many-cyclic system) the following rule can be used for determining the parity of two Kekulé structures: I f the Sachs graph obtained by superposition of two Kekulé graphs contains an even (odd) number of Am-membered rings, the Kekulé structures in question have the same (opposite) parity. Kekulé graphs depict Kekulé structures [62]. The Kekulé structure count is defined as,
K = K * + K - (53)
whilst the algebraic structure count [63] (corrected structure count [64]) ASC, is given by
ASC = K + - K ~ , K * > K ~ . (54)
and K ~ are related to the determinant of the adjacency matrix in the following way [58],
det A = (-l)^ /» (iir+ - JT )» + ( -1 ) ^ Y , (-1)«<»)2^*) (55) e€SJ,
where Sff is the set of Sachs graphs which are simultaneously spanning subgraphs of a graph and which contain at least one odd-membered cycle. (A spanning subgraph of G is a subgraph containing all the vertices of G [17].) For alternant hydrocarbons which do not contain odd-membered rings [19], S-N = 0 the above equation reduces to the Dewar-Longuet-Hig^ns formula [57],
detA = ( - l ) ^ /* ( J i r+ - J Í - ) . (56)
The application of formulae (55) and (56) will be illustrated in Figure 3.
