Arc Connectivity Research Articles

Erratum to: Arc Connectivity and Submodular Flows in Digraphs

Erratum to: Arc Connectivity and Submodular Flows in Digraphs

Arc Connectivity and Submodular Flows in Digraphs

Let D=(V,A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=(V,A)$$\\end{document} be a digraph. For an integer k≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 1$$\\end{document}, a k-arc-connected flip is an arc subset of D such that after reversing the arcs in it the digraph becomes (strongly) k-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a k-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer τ≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au \\ge 1$$\\end{document}, suppose dA+(U)+(τk-1)dA-(U)≥τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_A^+(U)+(\\frac{\ au }{k}-1)d_A^-(U)\\ge \ au $$\\end{document} for all U⊊V,U≠∅\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U\\subsetneq V, U\ e \\emptyset $$\\end{document}, where dA+(U)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_A^+(U)$$\\end{document} and dA-(U)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_A^-(U)$$\\end{document} denote the number of arcs in A leaving and entering U, respectively. Let C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document} be a crossing family over ground set V, and let f:C→Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:{\\mathcal {C}}\\rightarrow {\\mathbb {Z}}$$\\end{document} be a crossing submodular function such that f(U)≥kτ(dA+(U)-dA-(U))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(U)\\ge \\frac{k}{\ au }(d_A^+(U)-d_A^-(U))$$\\end{document} for all U∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U\\in {\\mathcal {C}}$$\\end{document}. Then D has a k-arc-connected flip J such that f(U)≥dJ+(U)-dJ-(U)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(U)\\ge d_J^+(U)-d_J^-(U)$$\\end{document} for all U∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U\\in {\\mathcal {C}}$$\\end{document}. The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams’ so-called weak orientation theorem, and proves a weaker variant of Woodall’s conjecture on digraphs whose underlying undirected graph is τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral.

Multiplicative topological semimetals

Exhaustive study of topological semimetal (TSM) phases of matter in equilibriated electonic systems and myriad extensions has built upon the foundations laid by earlier introduction and study of Weyl semimetals, with broad applications in topologically protected quantum computing, spintronics, and optical devices. We extend recent introduction of multiplicative topological phases to find previously overlooked TSM phases of electronic systems in equilibrium, with minimal symmetry protection. We show these multiplicative TSM (MTSM) phases exhibit rich and distinctive bulk-boundary correspondence and response signatures that greatly expand understanding of consequences of topology in condensed matter settings, such as the limits on Fermi arc connectivity and structure, and transport signatures such as the chiral anomaly. In this paper, we therefore lay the foundation for extensive future study of MTSMs. Published by the American Physical Society 2024

Static Evaluation of a Midimew Connected Torus Network for Next Generation Supercomputers

Many artificially intelligent systems solve complex health- and agriculture-related problems that require great computational power. Such systems are used for tracking medical records, genome sequence analysis, image-based plant disease detection, food supply chain traceability, and photosynthesis simulation. Massively parallel computers (MPCs) are among those used to solve these computation-intensive problems. MPCs comprise a million nodes; connecting such a large number of nodes is a daunting task. Therefore, hierarchical interconnection networks (HINs) have been introduced to solve this problem. A midimew-connected torus network (MTN) is a HIN that has basic modules (BM) as torus networks that are connected hierarchically by midimew links. This paper presents the performance of MTNs in terms of static topological parameters and cost-effectiveness, as measured through simulations. An MTN was compared with other networks, including mesh, torus, TESH, TTN, MMN, and TFBN. The results showed that our MTN had a low diameter with a high bisection width and arc connectivity. In addition, our MTN had a high cost–performance trade-off factor (CPTF), a high cost-effective factor (CEF), low packing density, and moderate message-traffic density with marginally higher costs, as compared to other networks, due to wire complexity. However, our MTN provided better bandwidth with higher static fault tolerance. Therefore, MTNs are suggested for further evaluation of the effective implementation of MPCs.

On the optimality of 3-restricted arc connectivity for digraphs and bipartite digraphs

On the optimality of 3-restricted arc connectivity for digraphs and bipartite digraphs

Expansive open Fermi arcs and connectivity changes induced by infrared phonons in ZrTe5

Expansive open Fermi arcs covering most of the surface Brillouin zone (SBZ) are desirable for detection and control of many topological phenomena, but they have generally been reported for Kramers-Weyl points, or unconventional chiral fermions, pinned at time-reversal invariant momentum in chiral materials. Here using first-principles band structure calculations, we show that for conventional Weyl points in $\mathrm{Zr}{\mathrm{Te}}_{5}$ with the chirality of +1/\ensuremath{-}1 near the BZ center at general momentum induced by one of the infrared phonons---the second lowest ${B}_{1u}$ mode for breaking inversion symmetry---they can also form expansive open Fermi arcs across the SBZ boundary to occupy most of the SBZ when projected on the (001) surface. We reveal that such expansive open Fermi arcs are evolved from the topological surface states that connect multiple surface Dirac points on the (001) surface of the topological insulator phases without lattice distortion in $\mathrm{Zr}{\mathrm{Te}}_{5}$. Furthermore, we find that the connectivity of the induced open Fermi arcs can be changed by the magnitude of the lattice distortion of this infrared phonon mode. Thus, we propose that using coherent optical phonons to modulate lattice parameters can offer ways to induce unique topological features including expansive open Fermi arcs and to dynamically control Fermi arc connectivity in $\mathrm{Zr}{\mathrm{Te}}_{5}$.

Midimew Connected Torus Network for Next Generation Massively Parallel Computer System

Important scientific and engineering applications need high performance computing. This can be provided by massively parallel computer (MPC) systems. A sensitive step of maintaining such systems is the interconnection network used to interconnect the computing nodes. The topology used effects the network costs and performance significantly. Hierarchal interconnection networks (HIN) were introduced having several attractive features including low latency, low cost, and high fault tolerance. This paper proposes a new HIN called Midimew connect Torus Network (MTN) that provides constant node degree, high arc connectivity, high fault tolerance and a reasonable bisection width. Static network performance evaluation for the proposed MTN has been conducted and compared with other networks. The comparison included conventional topologies such as 2D Mesh and 2D Torus and hierarchal ones which are TESH and TTN. The comparison of result shows that with the cost of extra communication links MTN attains higher fault tolerance than that of TESH and mesh networks, and equal to that of TTN, and less than that of torus network. Also, the hierarchical networks such as MTN, TTN, and TESH yield moderate bisection width; and the bisection width of MTN is lower than that of mesh and torus networks. This paper shows the superiority of the proposed MTN by comparative study of static network performance.

TFBN: A Cost Effective High Performance Hierarchical Interconnection Network

In order to fulfill the increasing demand for computation power to process a boundless data concurrently within a very short time or real-time in many areas such as IoT, AI, machine learning, smart grid, and big data analytics, we need exa-scale or zetta-scale computation in the near future. Thus, to have this level of computation, we need a massively parallel computer (MPC) system that shall consist of millions of nodes; and, for the interconnection of these massive numbers of nodes, conventional topologies are infeasible. Thus, a hierarchical interconnection network (HIN) is a rational way to connect huge nodes. Through this article, we are proposing a new HIN, which is a tori-connected flattened butterfly network (TFBN) for the next generation MPC system. Numerous basic modules are hierarchically interconnected as a toroidal connection, whereby the basic modules are flattened butterfly networks. We have studied the network architecture, static network performance, and static cost-effectiveness of the proposed TFBN in detail; and compared static network and cost-effectiveness performance of the TFBN to those of TTN, torus, TESH, and mesh networks. It is depicted that TFBN possesses low diameter and average distance, high arc connectivity, and temperate bisection width. It also has better cost-effectiveness and cost-performance trade-off factor compared to those of TTN, torus, TESH, and mesh networks. The only shortcoming is that the complexity of wiring of the TFBN is higher than that of those networks; this is because the basic module necessitates some extra short length link to form the flattened butterfly network. Therefore, TFBN is a high performance and cost-effective HIN, and it will be a good option for the next generation MPC system.

On the A α spectral radius of digraphs with given parameters

Let G be a digraph and be the adjacency matrix of G. Let be the diagonal matrix with outdegrees of vertices of G. For any real , define the matrix as The largest modulus of the eigenvalues of is called the spectral radius of G. In this paper, we determine the digraphs which attain the maximum (or minimum) spectral radius among all strongly connected digraphs with given parameters such as girth, clique number, vertex connectivity or arc connectivity. We also propose an open problem.

The (distance) signless Laplacian spectral radius of digraphs with given arc connectivity

Let G‾n,k denote the set of strongly connected digraphs with order n and arc connectivity k, and let G‾n,k⁎ denote the set of digraphs in G‾n,k with all vertices having outdegree and indegree greater than k. In this paper, we determine the unique digraph with the maximum signless Laplacian spectral radius among all digraphs in G‾n,k. We also determine the unique one with the maximum signless Laplacian spectral radius among all digraphs in G‾n,k⁎ with k=1,2. For the general case, we propose a conjecture on the maximum signless Laplacian spectral radius among all digraphs in G‾n,k⁎. Moreover, we characterize the extremal digraph achieving the minimum distance signless Laplacian spectral radius among all digraphs in G‾n,k. We also characterize the extremal digraph achieving the minimum distance signless Laplacian spectral radius among all digraphs in G‾n,k⁎ with k=1,2. For the general case, we propose a conjecture on the minimum distance signless Laplacian spectral radius among all digraphs in G‾n,k⁎.

Restricted Arc Connectivity of Unidirectional Hypercubes and Unidirectional Folded Hypercubes

Unidirectional hypercubes and unidirectional folded hypercubes are generalizations of hypercubes and folded hypercubes to digraphs. The super-$\lambda$ property of a digraph is a index for network reliability, which can be measured by the restricted arc-connectivity quantitatively. In this paper, we first show that the restricted arc-connectivity of the $n$-dimensional unidirectional hypercube is $n-1$ when $n$ is even and is $n-2$ when $n$ is odd, and then we show that the restricted arc-connectivity of the $n$-dimensional unidirectional folded hypercube is $n-1$ when $n$ is even and is $n$ when $n$ is odd. As a consequence, we prove that both unidirectional hypercube and unidirectional folded hypercube are super-$\lambda$.

The generalized distance matrix of digraphs

Let D(G) and DQ(G)=Diag(Tr)+D(G) be the distance matrix and distance signless Laplacian matrix of a simple strongly connected digraph G, respectively, where Diag(Tr)=diag(D1,D2,…,Dn) be the diagonal matrix with vertex transmissions of the digraph G. To track the gradual change of D(G) into DQ(G), in this paper, we propose to study the convex combinations of D(G) and Diag(Tr) defined byDα(G)=αDiag(Tr)+(1−α)D(G),0≤α≤1. This study reduces to merging the distance spectral and distance signless Laplacian spectral theories. The eigenvalue with the largest modulus of Dα(G) is called the Dα spectral radius of G, denoted by μα(G). We determine the digraph which attains the maximum (or minimum) Dα spectral radius among all strongly connected digraphs. Moreover, we also determine the digraphs which attain the minimum Dα spectral radius among all strongly connected digraphs with given parameters such as dichromatic number, vertex connectivity or arc connectivity.

Incidence cuts and connectivity in fuzzy incidence graphs

Fuzzy incidence graphs can be used as models for nondeterministic interconnection networks having extra node-edgerelationships. For example, ramps in a highway system may be modeled as a fuzzy incidence graph so that unexpectedflow between cities and highways can be effectively studied and controlled. Like node and edge connectivity in graphs,node connectivity and arc connectivity in fuzzy incidence graphs are introduced in this article. Their relationships withfuzzy connectivity parameters are discussed and results similar to Whitney’s theorems are obtained. Also, the incidenceis used to model flows in human trafficking networks.

Cartesian product digraphs with optimal restricted arc connectivity

The restricted arc connectivity of digraphs is a generalization of arc connectivity. In 2009, Chen and Liu [2] showed an upper bound on the restricted arc connectivity for Cartesian product digraphs. In this paper, we present a sufficient condition for this upper bound to be attained. Furthermore, we give an example to show that our result is best possible.

Differential Topology of Semimetals

The subtle interplay between local and global charges for topological semimetals exactly parallels that for singular vector fields. Part of this story is the relationship between cohomological semimetal invariants, Euler structures, and ambiguities in the torsion of manifolds. Dually, a topological semimetal can be represented by Euler chains from which its surface Fermi arc connectivity can be deduced. These dual pictures, and the link to topological invariants of insulators, are organised using geometric exact sequences. We go beyond Dirac-type Hamiltonians and introduce new classes of semimetals whose local charges are subtle Atiyah-Dupont-Thomas invariants globally constrained by the Kervaire semicharacteristic, leading to the prediction of torsion Fermi arcs.

3-Restricted arc connectivity of digraphs

The k-restricted arc connectivity of digraphs is a common generalization of the arc connectivity and the restricted arc connectivity. An arc subset S of a strong digraph D is a k-restricted arc cut if D−S has a strong component D′ with order at least k such that D−V(D′) contains a connected subdigraph with order at least k. The k-restricted arc connectivity λk(D) of a digraph D is the minimum cardinality over all k-restricted arc cuts of D.Let D be a strong digraph with order n≥6 and minimum degree δ(D). In this paper, we first show that λ3(D) exists if δ(D)≥3 and, furthermore, λ3(D)≤ξ3(D) if δ(D)≥4, where ξ3(D) is the minimum 3-degree of D. Next, we prove that λ3(D)=ξ3(D) if δ(D)≥n+32. Finally, we give examples showing that these results are best possible in some sense.

Arc fault tolerance of Kautz digraphs

The Kautz graphs are a famous class of network models and the super-λ property of graphs is a good indicator of the fault tolerance of networks. We, in this paper, study mainly the conditional arc connectivity and super-λ tolerance to arc-faults of digraphs, which are two measurements of super-λ property. We first give some bounds on them in terms of minimum degree of digraphs, and then determine them for Kautz digraphs. The obtained results show that the fault tolerance of Kautz networks is nice.

Surface Fermi arc connectivity in the type-II Weyl semimetal candidateWTe2

We perform ultrahigh resolution angle-resolved photoemission experiments at a temperature T=0.8 K on the type-II Weyl semimetal candidate WTe$_{2}$. We find a surface Fermi arc connecting the bulk electron and hole pockets on the (001) surface. Our results show that the surface Fermi arc connectivity to the bulk bands is strongly mediated by distinct surface resonances dispersing near the border of the surface-projected bulk band gap. By comparing the experimental results to first-principles calculations we argue that the coupling to these surface resonances, which are topologically trivial, is compatible with the classification of WTe$_{2}$ as a type-II Weyl semimetal hosting topological Fermi arcs. We further support our conclusion by a systematic characterization of the bulk and surface character of the different bands and discuss the similarity of our findings to the case of topological insulators.

Connectivity analysis and construction of cyclically balanced graphs

The concepts of connectivity and cyclic connectivity play an important role in weighted graph theory. The concepts of cyclic cutvertex, cyclic bridge and cyclically balanced graphs are discussed in this paper. It is proved that a vertex in a weighted graph is a cyclic cutvertex if it is a common vertex of all strong cycles with maximum strength. Also a cyclic cutvertex cannot be an endvertex in a weighted graph. A characterization of cyclically balanced graphs is obtained. Concepts of cyclic vertex connectivity and cyclic arc connectivity are also introduced.

Arc connectivity of transformation digraph Dxyz

A simple digraph D (without loops and multiple arcs)is said to be super-arc-connected if every minimum arc-cut is the out-neighbor set or in-neighbor set of a vertex. In this paper, we investigate the arc connectivity of these ten kinds of new total transformation digraphs associate with symbol ‘0’.