We list uses of, and the computational methods for the characteristic polynomial of a (chemical) graph. Pour computational methods are singled out for more detailed presentation. These are the graphical methods of Sachs, the recurrence formulae for several classes of simple graphs, the method based on Ulam subgraphs, and the Le Verrier — Faddeev — Franic recursive method. The latter method appears, at present, to be the most efficient procedure for the computation of the characteristic polynomials of graphs of sizes with up to even a few hundred sites.

The characteristic polynomial of a chemical graph

Electronic and structural properties of biazulene, terazulene, and polyazulene isomers

R. A. Davidson's rules for splitting an n-fold rotationally symmetric graph can be derived from a unitary transformation on the adjacency matrix. The McClelland and D'Amato rules are special cases with n = 2 and n = 3, respectively.

Splitting of n-fold rotationally symmetric graphs

The superhighway to greater understanding requires the development of techniques for facile methods (i.e. back of an envelope methods) of calculation and conceptualization of chemical phenomena. Huckel molecular orbital (HMO) theory is one of the vehicles on this superhighway. While reviewing some examples within this framework, new insights are given and select HMO data will be tabulated for the first time.

Techniques in facile calculation of molecular orbital parameters and related conceptualizations—molecular orbital functional groups

An algorithm has been developed for the construction of planar graphs for two series of IPR fullerenes of the general formula C50 + 10n and C60 + 12n
				(n = 1, 2, 3,…) maintaining 5- and 6-fold rotational symmetry, respectively. Some important utilities of these graphs have been shown. Simple rules have then been developed for factorising these graphs in order to obtain the eigenvalues corresponding to the totally symmetric irreducible representations of the C5 and C6 subgroups of the full point groups of fullerenes.

Construction of planar graphs for IPR fullerenes using 5- and 6-fold rotational symmetry: some eigenspectral analysis

It is pointed out that the method of McClelland [3] for the graphical factorization of Huckel matrices of conjugated systems can also be applied if the system has twofold symmetry but no plane of symmetry perpendicular to the molecular plane.

B.J. McClelland

Papers

On the graphical factorization of Hückel characteristic equations