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arXiv · 1302.5932

Some physical and chemical indices of clique-inserted-lattices

Zuhe Zhang
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Abstract

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.

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Zuhe Zhang. 2013-10-13. Some physical and chemical indices of clique-inserted-lattices. https://doi.org/10.1088/1742-5468%2F2013%2F10%2Fp10004

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