RESUMEN
In the tight-binding source and sink potential model of transmission in single-molecule pi-conjugated conductors, vanishing of the opacity polynomial defines a necessary condition for zero conductance at a given energy. Theorems are given for calculating opacity polynomials of composite devices in terms of opacity and characteristic polynomials of the subunits. These relations rationalize the positions and shapes of zeros in transmission curves for devices consisting of molecules with side chains or of units assembled in series and take an especially simple form for polymeric molecules with identical repeat units.
RESUMEN
We propose characterizing the cyclicity of molecular graphs by considering their D/DD matrix. Each nondiagonal element of D/DD is a quotient of the corresponding elements of the distance matrix D and the detour matrix DD of a graph. In particular, we are using the leading eigenvalue of the D/DD matrix as a descriptor of cyclicity and are investigating for monocyclic graphs Cn how this eigenvalue depends on the number of vertexes n, as n approaches infinity.
RESUMEN
Two definitions of the problem of graph drawing are considered, and an analytical solution is provided for each of them. The solutions obtained make use of the eigenvectors of the Laplacian matrix of a related structure. The procedures give good results for symmetrical graphs, and they have already been used for drawing fullerene molecules in the literature. The analysis characterizes precisely what problems the two procedures are solving. It also illuminates why they can perform unsatisfactorily on asymmetrical graphs.