Some physical and chemical indices of clique-inserted-lattices

The operation of replacing every vertex of an r-regular lattice H by a complete graph of order r is called clique-inserting, and the resulting lattice is called the clique-inserted-lattice of H. For any given r-regular lattice, applying this operation iteratively, an infinite family of r-regular lattices is generated. Some interesting lattices including the 3-12-12 lattice can be constructed this way. In this paper, we reveal the relationship between the energy and resistance distance of an r-regular lattice and that of its clique-inserted-lattice. As an application, the asymptotic energy per vertex and average resistance distance of the 3-12-12 and 3-6-24 lattices are computed. We also give formulae expressing the numbers of spanning trees and dimers of the k-th iterated clique-inserted lattices in terms of that of the original lattice. Moreover, we show that new families of expander graphs can be constructed from the known ones by clique-inserting.