Some Results On The Triangular Kagomé Lattices

The triangular Kagomé lattice (TKL for short), which is a new two dimensional latticesystem, has been studied extensively by combinatorialists and statistic physicists.The outline of this paper is as follows:In chapter one, we introduce some basic concepts of graph theory and some previousresults related to our topic. Toroidal triangular Kagomé lattice can be regarded as thesecond iterated line graph of the subdivision of the toroidal hexagonal lattice. In this paper,based on the relationship between the toroidal triangular Kagomé lattice and the toroidalhexagonal lattice, we deduce the eigenvalues as well as the Laplacian eigenvalues of thetoroidal triangular Kagomé lattice.In chapter two, we mainly introduce the spectrum and Laplacian spectrum of thetoroidal triangular Kagomé lattice.In chapter three, we derive the main results of our paper: the number of spanningtrees of the cylindrical triangular Kagomé lattice and the toroidal triangular Kagomé lat-tice, asymptotic formulas and asymptotic constants of the Kirchhof index as well as sometopological indices, such as energy, the number of perfect matching and so on.