Some physical and chemical indices of clique-inserted-lattices

TL;DR

This paper investigates the properties of clique-inserted lattices, revealing relationships between their energy, resistance distance, and spanning structures, and demonstrates how to construct new expander graphs through clique-inserting.

Contribution

It introduces formulas linking the energy, resistance distance, spanning trees, and dimers of clique-inserted lattices to the original lattice, and shows how to generate new expander graphs.

Findings

Derived formulas for energy and resistance distance of clique-inserted lattices.

Computed asymptotic energy per vertex and resistance distance for specific lattices.

Established methods to construct new expander graphs from known ones.

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.