Determining dimensionalities and multiplicities of crystal nets

Low-dimensional materials have attracted significant attention over the past decade. To discover new low-dimensional materials, high-throughput screening methods for structures with target dimensionality have been applied in different materials databases. For this purpose, the reliability of dimensionality identification is therefore highly important. In this work, we find that the existence of self-penetrating nets may lead to incorrect results by previous methods. Instead of this, we use the quotient graph to analyse the topologies of structures and compute their dimensionalities. Based on the quotient graph, we can calculate not only the dimensionality but also the multiplicity of self-penetrating structures. As a demonstration, we screened the Crystallography Open Database using the method and find hundreds of structures with different dimensionalities and high multiplicities up to 11. Some of the self-penetrating materials may have application values in gas storage, selective catalysis or photocatalysis because of their high gas sorption capacities and various electronic structures.

Affine Equivalence and Saddle Connection Graphs of Half-Translation Surfaces

To every half-translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces. We also investigate the automorphism group of the saddle connection graph and the corresponding quotient graph.

A general method to obtain the spectrum and local spectra of a graph from its regular partitions

It is well known that, in general, part of the spectrum of a graph can be obtained from the adjacency matrix of its quotient graph given by a regular partition. In this paper, a method that gives all the spectrum, and also the local spectra, of a graph from the quotient matrices of some of its regular partitions, is proposed. Moreover, from such partitions, the $C$-local multiplicities of any class of vertices $C$ is also determined, and some applications of these parameters in the characterization of completely regular codes and their inner distributions are described. As examples, it is shown how to find the eigenvalues and (local) multiplicities of walk-regular, distance-regular, and distance-biregular graphs.

Isotopy classes for 3-periodic net embeddings

Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. Periodic isotopy classifications are obtained for various families of embedded nets with small quotient graphs. The 25 periodic isotopy classes of depth-1 embedded nets with a single-vertex quotient graph are enumerated. Additionally, a classification is given of embeddings of n-fold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, as well as demonstrations of their maximal symmetry periodic isotopes. The methodology of linear graph knots on the flat 3-torus [0,1)3 is introduced. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases.

On the Vertex Multiplication Graphs

For any graph , with vertex set { } and a p-tuble of positive integers , the vertex multiplication graph is defined as the graph with vertex set consists of copies of each , where the copies of and are adjacent in if and only if the corresponding vertices and are adjacent in G . In this paper, we prove that the spectrum of is same as that of spectrum of its quotient graph with additional zero eigenvalues with multiplicity , where . Also we prove that the determinant of is minimum for and maximum for . Also we find distance- i spectrum of thorn graphs, , when G is connected - regular graph with diameter 2.

Graph polynomials and symmetries

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.

Vertex collisions in 3-periodic nets of genus 4

Unstable nets, by definition, display vertex collisions in any barycentric representation, among which are approximate models for the associated crystal structures. This means that different vertex lattices happen to superimpose when every vertex of a periodic net is located at the centre of gravity of its first neighbours. Non-crystallographic nets are known to be unstable, but crystallographic nets can also be unstable and general conditions for instability are not known. Moreover, examples of unstable nets are still scarce. This article presents a systematic analysis of unstable 3-periodic nets of genus 4, satisfying the restrictions that, in a suitable basis, (i) their labelled quotient graph contains a spanning tree with zero voltage and (ii) voltage coordinates belong to the set {−1, 0, 1}. These nets have been defined by a unique circuit of null voltage in the quotient graph. They have been characterized through a shortest path between colliding vertices. The quotient graph and the nature of the net obtained after identification of colliding vertices, if known, are also provided. The complete list of the respective unstable nets, with a detailed description of the results, can be found in the supporting information.