Regular Graphs Research Articles

The distinguishing index of graphs with infinite minimum degree

AbstractThe distinguishing index of a graph is the least number of colors necessary to obtain an edge coloring of that is preserved only by the trivial automorphism. We show that if is a connected ‐regular graph for some infinite cardinal then , proving a conjecture of Lehner, Pilśniak, and Stawiski. We also show that if is a graph with infinite minimum degree and at most vertices of degree for every infinite cardinal , then . In particular, if has infinite minimum degree and order at most .

The power of small coalitions under two-tier majority on regular graphs

In this paper, we study the following problem. Consider a setting where a proposal is offered to the vertices of a given network G, and the vertices must conduct a vote and decide whether to accept the proposal or reject it. Each vertex v has its own valuation of the proposal; we say that v is “happy” if its valuation is positive (i.e., it expects to gain from adopting the proposal) and “sad” if its valuation is negative. However, vertices do not base their vote merely on their own valuation. Rather, a vertex v is a proponent of the proposal if the majority of its neighbors are happy with it and an opponent in the opposite case. At the end of the vote, the network collectively accepts the proposal whenever the majority of its vertices are proponents. We study this problem for regular graphs with loops. Specifically, we consider the class Gn|d|h of d-regular graphs of odd order n with all n loops and h happy vertices. We are interested in establishing necessary and sufficient conditions for the class Gn|d|h to contain a labeled graph accepting the proposal, as well as conditions to contain a graph rejecting the proposal. We also discuss connections to the existing literature, including that on majority domination, and investigate the properties of the obtained conditions.

Graph Rigidity Properties of Ramanujan Graphs

A recent result of Cioabă, Dewar and Gu implies that any $k$-regular Ramanujan graph with $k \geq 8$ is globally rigid in $\mathbb{R}^2$. In this paper, we extend these results and prove that any $k$-regular Ramanujan graph of sufficiently large order is globally rigid in $\mathbb{R}^2$ when $k\in \{6, 7\}$, and when $k\in \{4,5\}$ if it is also vertex-transitive. These results imply that the Ramanujan graphs constructed by Morgenstern in 1994 are globally rigid. We also prove several results on other types of framework rigidity, including body-bar rigidity, body-hinge rigidity, and rigidity on surfaces of revolution. In addition, we use computational methods to determine which Ramanujan graphs of small order are globally rigid in $\mathbb{R}^2$.

New example of strongly regular graph with parameters [formula omitted] and a simple group [formula omitted] as the automorphism group

A new strongly regular graph with parameters (81,30,9,12) is found as a graph invariant under certain subgroup of the full automorphism group of the previously known strongly regular graph discovered in 1981 by J. H. van Lint and A. Schrijver.

Shared and individual representation learning with Feature Diversity for Deep MultiView Clustering

Due to the remarkable representation ability of Nonnegative matrix factorization (NMF), its multiview variants have become a crucial kind of multiview representation learning methods. However, the majority of existing methods fail to effectively utilize complex higher-level hierarchical information, shared information, and individual information of multiview data. To address these challenges, we propose the Shared and individual representation learning with feature Diversity for Deep Multiview Clustering (SiDDMVC). Specifically, this method decomposes data of each view through nonnegative matrix factorization in a layer-wise fashion to exploit complex higher-level hierarchical information. The outputs of NMF are then divided into shared and individual representations with shared geometrical constraint and feature diversity constraint. Experimental results on six real-world datasets have shown that SiDDMVC not only achieves superior performance through the use of graph regularization and two diversity constraints, but also surpasses several state-of-the-art multiview clustering methods.

Sampling frequency thresholds for the quantum advantage of the quantum approximate optimization algorithm

We compare the performance of the Quantum Approximate Optimization Algorithm (QAOA) with state-of-the-art classical solvers Gurobi and MQLib to solve the MaxCut problem on 3-regular graphs. We identify the minimum noiseless sampling frequency and depth p required for a quantum device to outperform classical algorithms. There is potential for quantum advantage on hundreds of qubits and moderate depth with a sampling frequency of 10 kHz. We observe, however, that classical heuristic solvers are capable of producing high-quality approximate solutions in linear time complexity. In order to match this quality for large graph sizes N, a quantum device must support depth p > 11. Additionally, multi-shot QAOA is not efficient on large graphs, indicating that QAOA p ≤ 11 does not scale with N. These results limit achieving quantum advantage for QAOA MaxCut on 3-regular graphs. Other problems, such as different graphs, weighted MaxCut, and 3-SAT, may be better suited for achieving quantum advantage on near-term quantum devices.

Boundary regularity of minimal graphs in the hyperbolic space

Abstract F.-H. Lin (1989) studied minimal graphs of the Dirichlet problem in the hyperbolic space and proved that any such minimal graph has the same global regularity as the boundary if the dimension of the minimal graph is even and that there is an obstacle to the higher regularity if the dimension is odd. We discuss the odd-dimensional case and study how the higher regularity is obstructed. We introduce the logarithmic distance to the boundary as an additional independent self-variable and establish concise boundary regularity for the minimal graph.

Generalized Eccentricity k^th Power Sum Adjacency Energy of Graphs (EGE^k SA(G))

Let be a finite, simple and undirected graph. For any integer , generalized eccentricity power sum adjacency matrix of is matrix with its entry as and zero otherwise, where is the eccentricity of the vertex v of a graph G. In this paper, the new energy of graph under the name as generalized eccentricity power sum adjacency energy of has been introduced. Generalized eccentricity power sum adjacency energy of some standard graphs and regular graphs obtained by complete graph.
 Let be a finite, simple and undirected graph. For any integer , generalized eccentricity power sum adjacency matrix of is matrix with its entry as and zero otherwise, where is the eccentricity of the vertex v of a graph G. In this paper, the new energy of graph under the name as generalized eccentricity power sum adjacency energy of has been introduced. Generalized eccentricity power sum adjacency energy of some standard graphs and regular graphs obtained by complete graph.

Results about the total chromatic number and the conformability of some families of circulant graphs

In this paper, we show that with the exception of the graph C12(1,3), which we prove to be Type 2, all members of the following three infinite families of 4-regular circulant graphs: Cn(2k,3), k≥1 and n=(8μ+6λ)k, for nonnegative integers μ and λ; C3n(1,3), for n≥3 and n≠4; and C3λp(1,p), for λ≥1 and p multiple of 3 are Type 1. The last two results are what is expected whether the Khennoufa and Togni’s conjecture is true, which states that 4-regular circulant graphs Cn(1,k) are Type 1 with 1<k<n2, but a finite number of Type 2 graphs. It is known that if a graph G is Type 1, then it is conformable. Furthermore, we introduce a relationship between the conformability problem and the equitable vertex coloring problem, by showing infinite families for which any equitable (Δ+1)-vertex coloring is conformable. In this context, we exhibit the infinite conformable graph family C(2q+1)n(d1,…,dq), n,q positive integers, containing C5n(d1,d2), which in turns, comprises one fifth of all 4-regular circulant graphs.

GFBLS: Graph-regularized fuzzy broad learning system for detection of interictal epileptic discharges

Epilepsy as the most common neurological disorder globally has drawn more and more attention. However, it is time-consuming and labor-intensive for manual detection of interictal epileptic discharges (IEDs). Thus, there is an urgent need to develop an efficient and automated detection approach as a more accurate diagnostic alternative for epilepsy detection. Recently, fuzzy broad learning system (FBLS) has been recognized as an alternative to deep learning and utilized in various fields. Nevertheless, FBLS ignores the locally invariant property of data. To effectively address this issue and further improve the performance of FBLS, a novel graph-regularized fuzzy broad learning system (GFBLS) is first proposed based on graph regularization. Moreover, an automated GFBLS-based approach is proposed for IEDs detection from EEG recordings. In the proposed method, graph convolutional neural networks (GCN) is firstly utilized to extract features from line graphs with undirected connections, which are constructed by EEG recordings, then extracted features by GCN are fed into GFBLS for IEDs detection. The experimental results demonstrated that GFBLS can achieve accuracy of 92.20%, specificity of 90.90% and precision of 91.13% with the training time of only 31.6 s, which is superior or comparable performance compared with other state-of-the-art approaches.

Complexity Phase Transitions Generated by Entanglement.

Entanglement is one of the physical properties of quantum systems responsible for the computational hardness of simulating quantum systems. But while the runtime of specific algorithms, notably tensor network algorithms, explicitly depends on the amount of entanglement in the system, it is unknown whether this connection runs deeper and entanglement can also cause inherent, algorithm-independent complexity. In this Letter, we quantitatively connect the entanglement present in certain quantum systems to the computational complexity of simulating those systems. Moreover, we completely characterize the entanglement and complexity as a function of a system parameter. Specifically, we consider the task of simulating single-qubit measurements of k-regular graph states on n qubits. We show that, as the regularity parameter is increased from 1 to n-1, there is a sharp transition from an easy regime with low entanglement to a hard regime with high entanglement at k=3, and a transition back to easy and low entanglement at k=n-3. As a key technical result, we prove a duality for the simulation complexity of regular graph states between low and high regularity.

A degree preserving delta wye transformation with applications to 6-regular graphs and Feynman periods

We investigate a degree preserving variant of the \Delta -Y transformation which replaces a triangle with a new 6-valent vertex which has double edges to the vertices that had been in the triangle. This operation is relevant for understanding scalar Feynman integrals in 6 dimensions. We study the structure of equivalence classes under this operation and its inverse, with particular attention to when the equivalence classes are finite, when they contain simple 6-regular graphs, and when they contain doubled 3-regular graphs. The last of these, in particular, is relevant for the Feynman integral calculations, and we make some observations linking the structure of these classes to the Feynman periods. Furthermore, we investigate properties of minimal graphs in these equivalence classes.

Ergodicity-to-localization transition on random regular graphs with large connectivity and in many-body quantum dots

Ergodicity-to-localization transition on random regular graphs with large connectivity and in many-body quantum dots

On Some Non-Rigid Unit Distance Patterns

A recent generalization of the Erdős Unit Distance Problem, proposed by Palsson, Senger, and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in 3-space. Studying a variant of this question, we prove sharp bounds on the number of unit distance paths and cycles on the sphere of radius 1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/{\\sqrt{2}}$$\\end{document}. We also consider a similar problem about 3-regular unit distance graphs in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}.

Some Inequalities over the Eigenvalues of a Strongly Regular Graph

Let’s consider a primitive strongly regular graph G. In this paper, we establish some inequalities over the spectrum of G in the environment of a real finite dimensional Euclidean Jordan algebra A associated with G recurring to a spectral analysis of some elements of A and recurring to a spectral analysis of the Generalized Krein parameters of G.

Random perfect matchings in regular graphs

AbstractWe prove that in all regular robust expanders , every edge is asymptotically equally likely contained in a uniformly chosen perfect matching . We also show that given any fixed matching or spanning regular graph in , the random variable is approximately Poisson distributed. This in particular confirms a conjecture and a question due to Spiro and Surya, and complements results due to Kahn and Kim who proved that in a regular graph every vertex is asymptotically equally likely contained in a uniformly chosen matching. Our proofs rely on the switching method and the fact that simple random walks mix rapidly in robust expanders.

On the Locating Rainbow Connection Number of Trees and Regular Bipartite Graphs

Locating the rainbow connection number of graphs is a new mathematical concept that combines the concepts of the rainbow vertex coloring and the partition dimension. In this research, we determine the lower and upper bounds of the locating rainbow connection number of a graph and provide the characterization of graphs with the locating rainbow connection number equal to its upper and lower bounds to restrict the upper and lower bounds of the locating rainbow connection number of a graph. We also found the locating rainbow connection number of trees and regular bipartite graphs. The method used in this study is a deductive method that begins with a literature study related to relevant previous research concepts and results, making hypotheses, conducting proofs, and drawing conclusions. This research concludes that only path graphs with orders 2, 3, 4, and complete graphs have a locating rainbow connection number equal to 2 and the order of graph G, respectively. We also showed that the locating rainbow connection number of bipartite regular graphs is in the range of r-⌊n/4⌋+2 to n/2+1, and the locating rainbow connection number of a tree is determined based on the maximum number of pendants or the maximum number of internal vertices. Doi: 10.28991/ESJ-2023-07-04-016 Full Text: PDF

Spectral mapping theorem of an abstract non-unitary quantum walk

This paper continues the previous work (Quantum Inf. Process 11 (2019)) by two authors of the present paper about a spectral mapping property of chiral symmetric unitary operators. In physics, they treat non-unitary time-evolution operators to consider quantum walks in open systems. In this paper, we generalize the above result to include a chiral symmetric non-unitary operator whose coin operator only has two eigenvalues. As a result, the spectra of such non-unitary operators are included in the (possibly non-unit) circle and the real axis in the complex plane. We also give some examples of our abstract results, such as non-unitary quantum walks defined by Mochizuki et al. Furthermore, the main theorem applies not only to quantum walks but also to the Ihara zeta function and correlated random walks on regular graphs.

The Application of Graph-Structured Cox Model in Financial Risk Early Warning of Companies

An effective financial risk forecast depends on the selection of important indicators from a broad set of financial indicators that are often correlated with one another. In this paper, we address this challenge by proposing a Cox model with a graph structure that allows us to identify and filter out the crucial indicators for financial risk forecasting. The Cox model can be converted to a weighted least squares form for the purpose of solution, where the regularization l0 compresses the signs of the variable coefficients and reduces the error caused by the compression of the coefficients. The graph structure reflects the correlations among different financial indicators and is incorporated into the model by introducing a Laplace penalty term to construct the Graph Regularization–Cox (GR-Cox) model. Monte Carlo simulation results show that the GR-Cox model outperforms the model without a graph structure with respect to the choice of parameters. Here, we apply the GR-Cox model to the forecast of the financial risk of listed companies and find that it shows good classification accuracy in practical applications. The GR-Cox model provides a new approach for improving the accuracy of financial risk early warning.

Adaptive unsupervised feature selection with robust graph regularization

Adaptive unsupervised feature selection with robust graph regularization