Graham, R. L. (ed.) et al., Handbook of combinatorics. Vol. 1-2. Amsterdam: Elsevier (North-Holland). 1955-1981 (1995).

Brief reference is made to some of the many important mathematical problems that still need to be solved in chemical combinatorics. Many of these problems are of interest to chemists and combinatorialists alike. Examples of such problems include the general solution of the hexagonal animal enumeration problem (which is equivalent to enumerating polycyclic aromatic hydrocarbons), and the characterization of the eigenvalue spectra, especially the occurrence of degeneracies, of the many polynomials used in bonding theory and other chemical applications. Such polynomials include the characteristic polynomial, the matching polynomial, the chromatic polynomial, the distance polynomial, random walk counting polynomials, the sextet polynomial, the permanental polynomial, and the polynomial associated with the Ising model partition function. The challenges of characterizing branching in molecular species, the computer perception of molecular symmetry, and the description of molecular similarity based on metric spaces are examples of other problems that need to be addressed. It would seem with all these problems (and the many others not mentioned here because of space limitations) that both chemists and combinatorialists will have more than enough to keep them occupied for at least another century. This brings to mind the words of the mathematician Sylvester first stated in 1878: “There is a wealth of untapped mathematical potential contained in the patient and long investigations of our chemist fellows”.

##### MSC:

92E10 | Molecular structures |

05A99 | Classical combinatorial problems |

05C90 | Applications of graph theory |

92E20 | Classical flows, reactions, etc. |

05A15 | Exact enumeration problems, generating functions |