Regular Graphs Research Articles

A convergence time of Grover walk on regular graph to stationary state with constant inflow to every vertex

We consider a quantum walk model on a finite graph which has an interaction with the outside. Here a quantum walker from the outside penetrates the graph and also a quantum walker in the graph goes out to the outside at every time step. This dynamics of the quantum walk converges to a stationary state. In this paper, we estimate the speed of the convergence to the stationary state on the κ-regular graph with the uniform insertion of the inflow to the graph. We show that larger degree of the regular graph makes the convergence time of this quantum walk model slower under this set-up.

Quantum walk mixing is faster than classical on periodic lattices

The quantum mixing time is a critical factor affecting the efficiency of quantum sampling and algorithm performance. It refers to the minimum time required for a quantum walk to approach its limiting distribution closely and has implications across the areas of quantum computation. This work focuses on the continuous time quantum walk mixing on a regular graph, evolving according to the unitary map U=eiĀt, where the Hamiltonian Ā is the normalized adjacency matrix of the graph. In [Physical Review A 76, 042306 (2007).], Richter previously showed that this walk mixes in time O(ndlog(d)log(1/ϵ)) with O(log(d)log(1/ϵ)) intermediate measurements when the graph is the d−dimensional periodic lattice Zn×Zn×⋯×Zn. We extend this analysis to the periodic lattice L=Zn1×Zn2×⋯×Znd, relaxing the assumption that ni are identical. We provide two quantum walks on periodic lattices that achieve faster mixing compared to classical random walks: 1. A coordinate-wise quantum walk that mixes in O((∑i=1dni)log(d/ϵ)) time with O(dlog(d/ϵ)) measurements. 2. A continuous-time quantum walk with O(log(1/ϵ)) measurements that conjecturally mixes in O(∑i=1dni(log(n1))2log(1/ϵ)) time. Our results demonstrate a quadratic speedup over the classical mixing time O(dn12log(d/ϵ)) on the generalized periodic lattice L. We have provided analytical evidence and numerical simulations to support the conjectured faster mixing time of the continuous-time quantum walk algorithm. Making progress towards proving the general conjecture that quantum walks on regular graphs mix in O(δ−1/2log(N)log(1/ϵ)) time, where δ is the spectral gap and N is the number of vertices.

On conflict-free proper colourings of graphs without small degree vertices

A proper vertex colouring of a graph G is conflict-free if in the neighbourhood of every vertex some colour appears exactly once, while it is called h-conflict-free if there are at least h such colours for each vertex of G. The least numbers of colours in such colourings of G are denoted by χpcf(G) and χpcfh(G), respectively. The latter parameter may be regarded as a natural relaxation of the 2-chromatic number, χ2(G), i.e. the least number of colours in a proper colouring of the square of a given graph G. It is known that χpcfh(G) can be as large as (h+1)(Δ+1)≈Δ2 for graphs with maximum degree Δ and h very close to Δ. We provide several new upper bounds for these parameters for graphs with minimum degrees δ large enough and h of smaller order than δ. In particular, we show that χpcfh(G)⩽(1+o(1))Δ if δ≫ln⁡Δ and h≪δ, and that χpcf(G)⩽Δ+O(ln⁡Δ) for regular graphs. These are related with the conjecture of Caro, Petruševski and Škrekovski that χpcf(G)⩽Δ+1 for every connected graph G of maximum degree Δ⩾3, towards which they proved that χpcf(G)⩽⌊5Δ2⌋ if Δ⩾1.

Coloring circle arrangements: New 4-chromatic planar graphs

Felsner, Hurtado, Noy and Streinu (2000) conjectured that arrangement graphs of simple great-circle arrangements have chromatic number at most 3. Motivated by this conjecture, we study the colorability of arrangement graphs for different classes of (pseudo-)circle arrangements.In this paper the conjecture is verified for △-saturated pseudocircle arrangements, i.e., for arrangements where one color class of the 2-coloring of faces consists of triangles only, as well as for further classes of (pseudo-)circle arrangements. These results are complemented by a construction which maps △-saturated arrangements with a pentagonal face to arrangements with 4-chromatic 4-regular arrangement graphs. This corona construction has similarities with the crowning construction introduced by Koester (1985). Based on exhaustive experiments with small arrangements we propose three strengthenings of the original conjecture.We further investigate fractional colorings. It is shown that the arrangement graph of every arrangement A of pairwise intersecting pseudocircles is “close” to being 3-colorable. More precisely, the fractional chromatic number χf(A) of the arrangement graph is bounded from above by χf(A)≤3+O(1n), where n is the number of pseudocircles of A. Furthermore, we construct an infinite family of 4-edge-critical 4-regular planar graphs which are fractionally 3-colorable. This disproves a conjecture of Gimbel, Kündgen, Li, and Thomassen (2019).

Grundy number of corona product of some graphs

The Grundy number of a graph G is the maximum number k of colors used to color the vertices of G such that the coloring is proper and every vertex u colored with color i, 1 ≤ i ≤ k, is adjacent to i – 1 vertices colored with each color j, 1 ≤ j ≤ i − 1. In this paper we obtain the Grundy number of corona product of some graphs, denoted by G ◦ H. First, we consider the graph G be 2-regular graph and H be a cycle, complete bipartite, ladder graph and fan graph. Also we consider the graph G and H be a complete bipartite graphs, fan graphs.

Identification of DNA N4-methylcytosine Sites via Multiview Kernel Sparse Representation Model

Identifying DNA N4-methylcytosine (4mC) sites is of great significance in biological research, such as chromatin structure, DNA stability, DNA-protein interaction and controlling gene expression. However, the traditional sequencing technology to identify 4mC sites is very time-consuming. In order to detect 4mC sites, we develop a multi-view learning method for achieving more effectively via merging multiple feature spaces. Furthermore, we think about whether the multi-view learning method can improve the across species classification ability by fusing data of multiple species. In our study, we propose a multi-view Laplacian kernel sparse representation-based classifier, called MvLapKSRC-HSIC. First, we make use of three feature extraction methods (PSTNP, NCP, DPP) to extract the DNA sequence features. MvLapKSRC-HSIC uses a kernel sparse representation-based classifier with graph regularization. In order to maintain the independence between various views, we add a multi-view regularization term constructed by Hilbert-Schmidt independence criterion (HSIC). In the experiments, MvLapKSRC-HSIC is applied on six datasets, so as to compare with other popular methods in single species and cross-species experiments. All experimental results show that MvLapKSRC-HSIC is superior to other outstanding methods on both single species and cross-species. Importantly, MvLapKSRC-HSIC can identify a series of potential DNA 4mC sites, which have not yet been experimentally evaluate on multiple species and merit further research.

On the existence of graphs which can colour every regular graph

On the existence of graphs which can colour every regular graph

Multiview Clustering via Hypergraph Induced Semi-Supervised Symmetric Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) based multiview technique has been commonly used in multiview data clustering tasks. However, previous NMF based multiview clustering approaches fail to take advantage of a small amount of supervisory information to effectively improve the clustering performance, and are easily affected by the additional post-processing method in clustering tasks. To cope with these issues, a novel framework named multiview clustering via hypergraph induced semi-supervised symmetric NMF (MVCHSS) is proposed in this paper for multiview data clustering applications. Specifically, the proposed method has the following features: 1) a new multiview based hypergraph pairwise constraints propagation (MHPCP) algorithm is developed in MVCHSS to construct a set of informative similarity matrices, revealing the high-order relationships effectively and fully utilizing the limited pairwise constraint supervisory information among samples of each view data; 2) the obtained similarity matrices with much supervisory information are not only enforced into the symmetric NMF (SNMF) model, but also incorporated into the graph regularization for each view data; 3) the optimization problem of MVCHSS is formulated for multiview data clustering tasks to acquire a more discriminative clustering indicator matrix (or called consensus assignment matrix) without additional post-processing method. Moreover, the proof of convergence and the computational complexity for MVCHSS are presented. Extensive experiments on five multiview datasets demonstrate that the proposed MVCHSS framework outperforms several state-of-the-art multiview clustering methods.

NLRRC: A Novel Clustering Method of Jointing Non-Negative LRR and Random Walk Graph Regularized NMF for Single-Cell Type Identification.

The development of single-cell RNA sequencing (scRNA-seq) technology has opened up a new perspective for us to study disease mechanisms at the single cell level. Cell clustering reveals the natural grouping of cells, which is a vital step in scRNA-seq data analysis. However, the high noise and dropout of single-cell data pose numerous challenges to cell clustering. In this study, we propose a novel matrix factorization method named NLRRC for single-cell type identification. NLRRC joins non-negative low-rank representation (LRR) and random walk graph regularized NMF (RWNMFC) to accurately reveal the natural grouping of cells. Specifically, we find the lowest rank representation of single-cell samples by non-negative LRR to reduce the difficulty of analyzing high-dimensional samples and capture the global information of the samples. Meanwhile, by using random walk graph regularization (RWGR) and NMF, RWNMFC captures manifold structure and cluster information before generating a cluster allocation matrix. The cluster assignment matrix contains cluster labels, which can be used directly to get the clustering results. The performance of NLRRC is validated on simulated and real single-cell datasets. The results of the experiments illustrate that NLRRC has a significant advantage in single-cell type identification.

Graph regularization centrality

Graph regularization centrality

Albertson (Alb) spectral radii and Albertson (Alb) energies of graph operation.

The sum of the absolute eigenvalues of the adjacency matrix make up graph energy. The greatest absolute eigenvalue of the adjacency matrix is represented by the spectral radius of the graph. Both molecular computing and computer science have uses for graph energies and spectral radii. The Albertson (Alb) energies and spectral radii of generalized splitting and shadow graphs constructed on any regular graph is the main focus of this study. The only thing that may be disputed is the comparison of the (Alb) energies and (Alb) spectral radii of the newly formed graphs to those of the base graph. By concentrating on splitting and shadow graph, we compute new correlations between the Alb energies and spectral radius of the new graph and the prior graph.

On security of GIS systems with N-tier architecture and family of graph based ciphers

Discovery of q-regular tree description in terms of an infinite system of quadratic equations over finite field Fq had an impact on Computer Science, theory of graph based cryptographic algorithms in particular. It stimulates the development of new graph based stream ciphers. It turns out that such encryption instruments can be efficiently used in GIS protection systems which use N-Tier Architecture. We observe known encryption algorithms based on the approximations of regular tree, their modifications defined over arithmetical rings and implementations of these ciphers. Additionally new more secure graph based ciphers suitable for GIS protection will be presented.Algorithms are constructed using vertices of bipartite regular graphs D(n,K) defined by a finite commutative ring K with a unit and a non-trivial multiplicative group K*. The partition of such graphs are n-dimensional affine spaces over the ring K. A walk of even length determines the transformation of the transition from the initial to the last vertex from one of the partitions of the graph. Therefore, the affine space Kn can be considered as a space of plaintexts, and walking on the graph is a password that defines the encryption transformation.With certain restrictions on the password the effect when different passwords with K*)2s, s <[(n+5)/2]/2 correspond to different ciphertexts of the selected plaintext with Kn can be achieved.In 2005, such an algorithm in the case of the finite field F127 was implemented for the GIS protection. Since then, the properties of encryption algorithms using D(n, K) graphs (execution speed, mixing properties, degree and density of the polynomial encryption transform) have been thoroughly investigated.The complexity of linearization attacks was evaluated and modifications of these algorithms with the resistance to linearization attacks were found. It turned out that together with D(n, K) graphs, other algebraic graphs with similar properties, such as A(n, K) graphs, can be effectively used.The article considers several solutions to the problem of protecting the geological information system from possible cyberattacks using stream ciphers based on graphs. They have significant advantages compared to the implemented earlier systems.

Some snarks are worse than others

Many conjectures and open problems in graph theory can either be reduced to cubic graphs or are directly stated for them. Furthermore, it is known that for a lot of problems, a counterexample must be a snark, that is, a bridgeless cubic graph which is not 3-edge-colourable. In this paper we deal with the fact that the family of potential counterexamples to many interesting conjectures can be narrowed even further to the family S≥5 of bridgeless cubic graphs whose edge set cannot be covered with four perfect matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover Conjecture and the Fan–Raspaud Conjecture are examples of statements for which S≥5 is crucial. In this paper we study parameters which have the potential to further refine S≥5 and thus enlarge the set of cubic graphs for which the mentioned conjectures can be verified. We show that S≥5 can be naturally decomposed into subsets with increasing complexity, thereby producing a natural scale for proving these conjectures. More precisely, we consider the following parameters and questions: given a bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii) how many copies of the same perfect matching need to be added, and (iii) how many 2-factors need to be added so that the resulting regular graph is Class I? We present new results for these parameters and we also establish some strong relations between these problems and some long-standing conjectures.

The Subfield and Extended Codes of a Subclass of Optimal Three-Weight Cyclic Codes

A class of optimal three-weight [q^k-1,k+1,q^{k-1}(q-1)-1] cyclic codes over {mathrm{I!F}}_q, with kge 2, achieving the Griesmer bound, was presented by Heng and Yue (IEEE Trans Inf Theory 62(8):4501–4513, 2016. https://doi.org/10.1109/TIT.2016.2550029). In this paper we study some of the subfield codes of this class of optimal cyclic codes when k=2. The weight distributions of the subfield codes are settled. It turns out that some of these codes are optimal and others have the best known parameters. The duals of the subfield codes are also investigated and found to be almost optimal with respect to the sphere-packing bound. In addition, the covering structure for the studied subfield codes is determined. Some of these codes are found to have the important property that any nonzero codeword is minimal, which is a desirable property that is useful in the design of a secret sharing scheme based on a linear code. Moreover, a specific example of a secret sharing scheme based on one of these subfield codes is given. Finally, a class of optimal two-weight linear codes over {mathrm{I!F}}_q, achieving the Griesmer bound, whose duals are almost optimal with respect to the sphere-packing bound is presented. Through a different approach, this class of optimal two-weight linear codes was reported very recently by Heng (IEEE Trans Inf Theory 69(2):978–994, 2023. https://doi.org/10.1109/TIT.2022.3203380). Furthermore, it is shown that these optimal codes can be used to construct strongly regular graphs.

Elastic deep autoencoder for text embedding clustering by an improved graph regularization

Text clustering is a task for grouping extracted information of the text in different clusters, which has many applications in recommender systems, sentiment analysis, and more. Deep learning-based methods have become increasingly popular due to their high accuracy in identifying nonlinear structures. They usually consist of two major parts: dimensionality reduction and clustering. Autoencoders are simple unsupervised neural networks used for better representation of low-dimensional data and have shown good performance in dealing with non-linear features. However, while they utilize the Frobenius norm to deal well with Gaussian noise, they are sensitive to outlier data and Laplacian noise. In this paper, a deep autoencoder with an adapted elastic loss for text embedding clustering (EDA-TEC) is proposed. The elastic loss is a combination of the Frobenius norm and L2,1-norm to consider both types of noises. Additionally, to maintain the high-dimensional data geometric structure, a modified graph regularization term based on the weighted cosine similarity measure is used. EDA-TEC also improves clustering results by considering the sparsity regularization of the manifold representation data. In this jointly end-to-end deep learning model, better representation and text clustering results are achieved with high accuracy on common datasets compared to existing methods.11https://github.com/safinal/text-embedding-clustering.

On structural parameterizations of load coloring

Given a graph G and a positive integer k, the 2-Load Coloring problem is to check whether there is a 2-coloring f:V(G)→{r,b} of G such that for every i∈{r,b}, there are at least k edges with both end vertices colored i. It is known that the problem is NP-complete even on special classes of graphs like regular graphs. Gutin and Jones (2014) showed that the problem is fixed-parameter tractable by giving a kernel with at most 7k vertices. Barbero et al. (2017) obtained a kernel with less than 4k vertices and O(k) edges, improving the earlier result.In this paper, we study the parameterized complexity of the problem with respect to structural graph parameters. We show that 2-Load Coloring cannot be solved in time f(w)no(w), unless ETH fails and it can be solved in time nO(w), where n is the size of the input graph, w is the clique-width of the graph and f is an arbitrary function of w. Next, we consider the parameters distance to cluster graphs, distance to co-cluster graphs and distance to threshold graphs, which are weaker than the parameter clique-width and show that the problem is fixed-parameter tractable (FPT) with respect to these parameters. Finally, we show that 2-Load Coloring is NP-complete even on bipartite graphs and split graphs.

Several classes of linear codes with few weights over finite fields

Linear codes with few weights have applications in secret sharing schemes, authentication codes, association schemes and strongly regular graphs. In this paper, we introduce a new flexible defining set, and consequently, we obtain several classes of t-weight linear codes over Fp, where t=4,5,6 and p is an odd prime. The parameters and weight distributions of the generated linear codes are fully determined using quadratic Gauss sums over finite fields. Moreover, two classes of optimal two-weight codes meeting the Griesmer bound are obtained from our constructions. Our results are heavily based on evaluating some exponential sums over finite fields. Computer experiments using MAGMA programs show that many (almost) optimal codes can be derived from our results.

Optimally reconfiguring list and correspondence colourings

The reconfiguration graph Ck(G) for the k-colourings of a graph G has a vertex for each proper k-colouring of G, and two vertices of Ck(G) are adjacent precisely when those k-colourings differ on a single vertex of G. Much work has focused on bounding the maximum value of diamCk(G) over all n-vertex graphs G. We consider the analogous problems for list colourings and for correspondence colourings. We conjecture that if L is a list-assignment for a graph G with |L(v)|≥d(v)+2 for all v∈V(G), then diamCL(G)≤n(G)+μ(G). We also conjecture that if (L,H) is a correspondence cover for a graph G with |L(v)|≥d(v)+2 for all v∈V(G), then diamC(L,H)(G)≤n(G)+τ(G). (Here μ(G) and τ(G) denote the matching number and vertex cover number of G.) For every graph G, we give constructions showing that both conjectures are best possible, which also hints towards an exact form of Cereceda’s Conjecture for regular graphs. Our first main result proves the upper bounds (for the list and correspondence versions, respectively) diamCL(G)≤n(G)+2μ(G) and diamC(L,H)(G)≤n(G)+2τ(G). Our second main result proves that both conjectured bounds hold, whenever all v satisfy |L(v)|≥2d(v)+1. We conclude by proving one or both conjectures for various classes of graphs such as complete bipartite graphs, subcubic graphs, cactuses, and graphs with bounded maximum average degree. The full paper can also be found at arxiv.org/abs/2204.07928.

Large population limits of Markov processes on random networks

We consider time-continuous Markovian discrete-state dynamics on random networks of interacting agents and study the large population limit. The dynamics are projected onto low-dimensional collective variables given by the shares of each discrete state in the system, or in certain subsystems, and general conditions for the convergence of the collective variable dynamics to a mean-field ordinary differential equation are proved. We discuss the convergence to this mean-field limit for a continuous-time noisy version of the so-called “voter model” on Erdős–Rényi random graphs, on the stochastic block model, and on random regular graphs. Moreover, a heterogeneous population of agents is studied.

Some exact results for regular Turán problems for all large orders

As a variant of the famous Turán problem, we study rex(n,F), the maximum number of edges that an n-vertex regular graph can have without containing a copy of F. We determine rex(n,Kr+1) for all pairs of integers r and large enough n. For every tree T, we determine rex(n,T) for every n large enough.