Based on the total π-electron energies Eπs of Hückel Molecular Orbital (HMO) method for all the possible isomers of conjugated acyclic polyenes (C₂nH₂n+2) up to n = 7, the structure-stability relation of the possible isomers was analyzed. It was shown that the mean length of conjugation L can roughly predict the ordering of stability among isomers, while the Z-index, or Hosoya-index, can almost perfectly reproduce their stability. Further, the genealogy of the conjugated acyclic polyene family was obtained by drawing systematic diagrams connecting these isomers of different n, and governed by several simple rules. Namely, the stability change of a given isomer in the genealogy connecting n and n + 1 polyenes can be classified into three different modes of vinyl addition (elongation, inner and outer branching) and horn growing, i.e., substitution of -HC=CH- moiety with -HC(=CH₂)-C(=CH₂)H-. By using the Z-index, we can extend this type of discussion to polyene radicals and even to "cross-conjugated" cyclic polyenes containing only one odd-membered cycle, such as radialene and fulvene.
Keywords: Hosoya index; Kekulé structure; branching; conjugated polyene; cross-conjugation; mean length of conjugation; structure-stability relation; topological index; total π-electronic energy of HMO.