A topological index for the totalπ-electron energy

AbstractA modified topological index $$\tilde Z_G $$ is proposed to be defined as $$\tilde Z_G = \sum\limits_{k = 0}^{[N/2]} {( - 1)^k } a_{2k} $$ for characterising theπ-electronic system of a conjugated hydrocarbonG withN carbon atoms, wherea2k is the coefficient of the characteristic polynomial ofG defined as $$P_G (X) = ( - 1)^N \det |A - XE| = \sum\limits_{k = 0}^N { a_k X^{N - k} } $$ with an adjacency matrixA and the unit matrixE. $$\tilde Z_G $$ is identical toZG for a tree graph, or a chain hydrocarbon.ZG increases with a (4n+2)-membered ring formation and decreases with a 4n-membered ring formation. The totalπ-electron energyEπ of the Hückel molecular orbital is shown to be related with $$\tilde Z_G $$ asEπ =Cln $$\tilde Z_G $$ . With this relation generalised and extended Hückel rules for predicting the stability of an arbitrary network are proved.

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