Cubic Graphs Research Articles

The Local Metric Dimension and Distance-Edge-Monitoring Number of Graph

Let [Formula: see text] be a graph and [Formula: see text]. The vertex [Formula: see text] is said to resolve a pair [Formula: see text] and [Formula: see text] if and only if [Formula: see text]. A set [Formula: see text] is defined as a resolving set of [Formula: see text] if for all [Formula: see text], the pair [Formula: see text] is resolved by some [Formula: see text]. The minimum cardinality of a resolve set of [Formula: see text] is defined as [Formula: see text]. A set [Formula: see text] is a local resolve set of [Formula: see text] if for all [Formula: see text] such that [Formula: see text], the pair [Formula: see text] is resolved by some [Formula: see text]. The minimum cardinality of a local resolve set of [Formula: see text] is defined as [Formula: see text]. An edge [Formula: see text] is said to be monitored by [Formula: see text] if [Formula: see text] or [Formula: see text]. A set [Formula: see text] is a distance-edge-monitoring (DEM) set if for all [Formula: see text], [Formula: see text] is monitored by some [Formula: see text]. The minimum cardinality of a [Formula: see text] set of [Formula: see text] is defined as [Formula: see text]. In this paper, we obtained that [Formula: see text] for all non trivial graphs with order [Formula: see text], and the exact value of [Formula: see text] and [Formula: see text] for [Formula: see text], [Formula: see text]. Also, we obtained that if [Formula: see text] then [Formula: see text]. With respect to the relation between the defined graph invariants, it was proved a bound for [Formula: see text] for [Formula: see text] and the exact values of [Formula: see text] for [Formula: see text], where [Formula: see text] (resp, [Formula: see text])is the maximum value of [Formula: see text] (resp, [Formula: see text]) over all graphs [Formula: see text] with order [Formula: see text]. Finally, we proved that for [Formula: see text], there exists a graph with order [Formula: see text] such that [Formula: see text] and [Formula: see text].

Embedding of Hypercube into Fractal Cubic Network

The implementation of parallel algorithms and the simulation of interconnection networks can be modeled into a graph embedding problem. Several cost parameters are used to assess the quality of an embedding. The wirelength is one of these factors that is frequently taken into account. A fractal cubic network is a new variant of the hypercube graph. In this study, we compute the wirelength of an embedding of the hypercube [Formula: see text] into the fractal cubic network [Formula: see text] using [Formula: see text]-partition.

Large dilates of hypercube graphs in the plane

Large dilates of hypercube graphs in the plane

On Graphs Embeddable in a Layer of a Hypercube and Their Extremal Numbers

AbstractA graph is cubical if it is a subgraph of a hypercube. For a cubical graph H and a hypercube $$Q_n$$ Q n , $$\textrm{ex}(Q_n, H)$$ ex ( Q n , H ) is the largest number of edges in an H-free subgraph of $$Q_n$$ Q n . If $$\textrm{ex}(Q_n, H)$$ ex ( Q n , H ) is at least a positive proportion of the number of edges in $$Q_n$$ Q n , then H is said to have positive Turán density in the hypercube; otherwise it has zero Turán density. Determining $$\textrm{ex}(Q_n, H)$$ ex ( Q n , H ) and even identifying whether H has positive or zero Turán density remains a widely open question for general H. In this paper we focus on layered graphs, i.e., graphs that are contained in an edge layer of some hypercube. Graphs H that are not layered have positive Turán density because one can form an H-free subgraph of $$Q_n$$ Q n consisting of edges of every other layer. For example, a 4-cycle is not layered and has positive Turán density. However, in general, it is not obvious what properties layered graphs have. We give a characterization of layered graphs in terms of edge-colorings. We show that most non-trivial subdivisions have zero Turán density, extending known results on zero Turán density of even cycles of length at least 12 and of length 8. However, we prove that there are cubical graphs of girth 8 that are not layered and thus having positive Turán density. The cycle of length 10 remains the only cycle for which it is not known whether its Turán density is positive or not. We prove that $$\textrm{ex}(Q_n, C_{10})= \Omega (n2^n/ \log ^a n)$$ ex ( Q n , C 10 ) = Ω ( n 2 n / log a n ) , for a constant a, showing that the extremal number for a 10-cycle behaves differently from any other cycle of zero Turán density.

Edges deletion problem of hypercube graphs for some n

Numerous graph-theoretic methods can be used to investigate the efficiency and reliability of networks. The reliability of a network is determined based on its connectivity. At the very least, the system must continue to function in the event of part of its component (node or link) failures. An ideal system remains reliable even when errors occur. Diameter is a measure of network efficiency used to study the effects of link failures in networks. In this paper, we look into the eliminating edges affect to the hypercube graph’s diameter (call it by [Formula: see text]) and calculate the maximum diameter that is denoted by [Formula: see text] after deleting [Formula: see text] edges from hypercube graphs [Formula: see text] for some n.

On discrete constant principal curvature surfaces

It has been recently discovered that a certain class of nanocarbon materials has geometrical properties related to the geometry of discrete surfaces with a pre-constant discrete curvature, based on a discrete surface theory for trivalent graphs proposed in 2017 by Kotani et al. In this paper, with the aim of an application to the nanocarbon materials, we will study discrete constant principal curvature (CPC) surfaces. Firstly, we develop the discrete surface theory on a full 3-ary oriented tree so that we define a discrete analogue of principal directions on them and investigate it. We also construct some interesting examples of discrete constant principal curvature surfaces, including discrete CPC tori.

A Weierstrass Representation Formula for Discrete Harmonic Surfaces

A discrete harmonic surface is a trivalent graph which satisfies the balancing condition in the 3-dimensional Euclidean space and achieves energy minimizing under local deformations. Given a topological trivalent graph, a holomorphic function, and an associated discrete holomorphic quadratic form, a version of the Weierstrass representation formula for discrete harmonic surfaces in the 3-dimensional Euclidean space is proposed. By using the formula, a smooth converging sequence of discrete harmonic surfaces is constructed, and its limit is a classical minimal surface defined with the same holomorphic data. As an application, we have a discrete approximation of the Enneper surface.

Limited Query Graph Connectivity Test

We propose a combinatorial optimisation model called Limited Query Graph Connectivity Test. We consider a graph whose edges have two possible states (On/Off). The edges' states are hidden initially. We could query an edge to reveal its state. Given a source s and a destination t, we aim to test s−t connectivity by identifying either a path (consisting of only On edges) or a cut (consisting of only Off edges). We are limited to B queries, after which we stop regardless of whether graph connectivity is established. We aim to design a query policy that minimizes the expected number of queries. Our model is mainly motivated by a cyber security use case where we need to establish whether attack paths exist in a given network, between a source (i.e., a compromised user node) and a destination (i.e., a high-privilege admin node). Edge query is resolved by manual effort from the IT admin, which is the motivation behind query minimization. Our model is highly related to Stochastic Boolean Function Evaluation (SBFE). There are two existing exact algorithms for SBFE that are prohibitively expensive. We propose a signifcantly more scalable exact algorithm. While previous exact algorithms only scale for trivial graphs (i.e., past works experimented on at most 20 edges), we empirically demonstrate that our algorithm is scalable for a wide range of much larger practical graphs (i.e., graphs representing Windows domain networks with tens of thousands of edges). We also propose three heuristics. Our best-performing heuristic is via limiting the planning horizon of the exact algorithm. The other two are via reinforcement learning (RL) and Monte Carlo tree search (MCTS). We also derive an algorithm for computing the performance lower bound. Experimentally, we show that all our heuristics are near optimal. The heuristic building on the exact algorithm outperforms all other heuristics, surpassing RL, MCTS and eight existing heuristics ported from SBFE and related literature.

Guided quantum walk

We utilize the theory of local amplitude transfer (LAT) to gain insights into quantum walks (QWs) and quantum annealing (QA) beyond the adiabatic theorem. By representing the eigenspace of the problem Hamiltonian as a hypercube graph, we demonstrate that probability amplitude traverses the search space through a series of local Rabi oscillations. We argue that the amplitude movement can be systematically guided towards the ground state using a time-dependent hopping rate based solely on the problem's energy spectrum. Building upon these insights, we extend the concept of multistage QW by introducing the guided quantum walk (GQW) as a bridge between QW-like and QA-like procedures. We assess the performance of the GQW on exact cover, traveling salesperson, and garden optimization problems with 9 to 30 qubits. Our results provide evidence for the existence of optimal annealing schedules, beyond the requirement of adiabatic time evolutions. These schedules might be capable of solving large-scale combinatorial optimization problems within evolution times that scale linearly in the problem size. Published by the American Physical Society 2024

IDEMPOTENT GENERATORS OF INCIDENCE ALGEBRAS

Abstract The minimum number of idempotent generators is calculated for an incidence algebra of a finite poset over a commutative ring. This quantity equals either $\lceil \log _2 n\rceil $ or $\lceil \log _2 n\rceil +1$ , where n is the cardinality of the poset. The two cases are separated in terms of the embedding of the Hasse diagram of the poset into the complement of the hypercube graph.

Lower Bounds on the Homology of Vietoris–Rips Complexes of Hypercube Graphs

We provide novel lower bounds on the Betti numbers of Vietoris–Rips complexes of hypercube graphs of all dimensions and at all scales. In more detail, let Qn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_n$$\\end{document} be the vertex set of 2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^n$$\\end{document} vertices in the n-dimensional hypercube graph, equipped with the shortest path metric. Let VR(Qn;r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)$$\\end{document} be its Vietoris–Rips complex at scale parameter r≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r \\ge 0$$\\end{document}, which has Qn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_n$$\\end{document} as its vertex set, and all subsets of diameter at most r as its simplices. For integers r<r′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r<r'$$\\end{document} the inclusion VR(Qn;r)↪VR(Qn;r′)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)\\hookrightarrow \ extrm{VR}(Q_n;r')$$\\end{document} is nullhomotopic, meaning no persistent homology bars have length longer than one, and we therefore focus attention on the individual spaces VR(Qn;r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)$$\\end{document}. We provide lower bounds on the ranks of homology groups of VR(Qn;r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)$$\\end{document}. For example, using cross-polytopal generators, we prove that the rank of H2r-1(VR(Qn;r))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{2^r-1}(\ extrm{VR}(Q_n;r))$$\\end{document} is at least 2n-(r+1)nr+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{n-(r+1)}\\left( {\\begin{array}{c}n\\\\ r+1\\end{array}}\\right) $$\\end{document}. We also prove a version of homology propagation: if q≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\ge 1$$\\end{document} and if p is the smallest integer for which rankHq(VR(Qp;r))≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{rank}H_q(\ extrm{VR}(Q_p;r))\ e 0$$\\end{document}, then rankHq(VR(Qn;r))≥∑i=pn2i-pi-1p-1·rankHq(VR(Qp;r))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{rank}H_q(\ extrm{VR}(Q_n;r)) \\ge \\sum _{i=p}^n 2^{i-p} \\left( {\\begin{array}{c}i-1\\\\ p-1\\end{array}}\\right) \\cdot \ extrm{rank}H_q(\ extrm{VR}(Q_p;r))$$\\end{document} for all n≥p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge p$$\\end{document}. When r≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\le 3$$\\end{document}, this result and variants thereof provide tight lower bounds on the rank of Hq(VR(Qn;r))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_q(\ extrm{VR}(Q_n;r))$$\\end{document} for all n, and for each r≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r \\ge 4$$\\end{document} we produce novel lower bounds on the ranks of homology groups. Furthermore, we show that for each r≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge 2$$\\end{document}, the homology groups of VR(Qn;r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)$$\\end{document} for n≥2r+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 2r+1$$\\end{document} contain propagated homology not induced by the initial cross-polytopal generators.

Integer k-matching preclusion of some interconnection networks

For a positive integer k, a k-matching of a graph G is a function f: E(G)→{0,1,…,k} such that ∑e∈Γ(v)f(e)≤k for every vertex v of G, where Γ(v) represents the set of all edges incident to v. The (strong) k-matching preclusion number of G, denoted by (smpk(G))mpk(G), is the minimum size of (edges and vertices) edges whose deletion leaves the remaining subgraph that has neither perfect k-matchings nor almost perfect k-matchings. This is a generalization of the concept matching preclusion problem proposed by Brigham et al. In this paper, we consider the k-matching preclusion problem of some interconnection networks, such as the hypercube-like graphs HLn, the augmented cubes AQn and the torus networks Cm1□Cm2□⋯□Cmn, which are variants of the hypercube graphs Qn. Our result on restricted hypercube-like graphs RHLn generalizes the result of twisted cubes TQn by Chang et al. [3].

Differential graded algebras for trivalent plane graphs and their representations

To any trivalent plane graph embedded in the sphere Casals and Murphy associate a differential graded algebra (dg-algebra), in which the underlying graded algebra is free associative over a commutative ring. Our first result is the construction of a generalization of the Casals–Murphy dg-algebra to non-commutative coefficients, for which we prove various functoriality properties not previously verified in the commutative setting. Our second result is to prove that rank r representations of this dg-algebra, over a field \mathbb{F} , correspond to colorings of the faces of the graph by elements of the Grassmannian \operatorname{Gr}(r,2r;\mathbb{F}) so that bordering faces are transverse, up to the natural action of \operatorname{PGL}_{2r}(\mathbb{F}) . Underlying the combinatorics, the dg-algebra is a computation of the fully non-commutative Legendrian contact dg-algebra for Legendrian satellites of Legendrian 2-weaves, though we do not prove as such in this paper. The graph coloring problem verifies that, for Legendrian 2-weaves, rank r representations of the Legendrian contact dg-algebra correspond to constructible sheaves of microlocal rank r . This is the first explicit such computation of the bijection between the moduli spaces of representations and sheaves for an infinite family of Legendrian surfaces.

Tropical Fock–Goncharov coordinates for -webs on surfaces I: construction

Abstract For a finite-type surface $\mathfrak {S}$ , we study a preferred basis for the commutative algebra $\mathbb {C}[\mathscr {R}_{\mathrm {SL}_3(\mathbb {C})}(\mathfrak {S})]$ of regular functions on the $\mathrm {SL}_3(\mathbb {C})$ -character variety, introduced by Sikora–Westbury. These basis elements come from the trace functions associated to certain trivalent graphs embedded in the surface $\mathfrak {S}$ . We show that this basis can be naturally indexed by nonnegative integer coordinates, defined by Knutson–Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock and Goncharov, to the tropical points at infinity of the dual version of the character variety.

Irregular chromatic number for hypercube graph and its variants

The attractive properties of the hypercube graph such as its diameter, good connectivity, and symmetry have made it a popular topology for the design of multi-computer interconnection networks. Efforts to improve some of these properties have led to the evolution of hypercube variants. Let c be the proper coloring of graph G, where the neighboring vertices will get individual colors. Coloring c is irregular if distinct vertices have distinct color codes and the least number of colors that ought to receive an irregular coloring is the irregular chromatic number, χir (G). In this paper, we discuss the irregular coloring and find the irregular chromatic number for the hypercube graph Qn and some of its variants using binomial coefficients for the Locally twisted cube graph LTQn, Crossed cube graph CQn and two types of Fractal cubic network graph FCNG1 (k) and FCNG2 (k).

Quantum symmetries of Cayley graphs of abelian groups

AbstractWe study Cayley graphs of abelian groups from the perspective of quantum symmetries. We develop a general strategy for determining the quantum automorphism groups of such graphs. Applying this procedure, we find the quantum symmetries of the halved cube graph, the folded cube graph, and the Hamming graphs.

Density-Based Distance Preserving Graph: Theoretical and Practical Analyses.

This brief aims to provide theoretical guarantee and practical guidance on constructing a type of graphs from input data via distance preserving criterion. Unlike the graphs constructed by other methods, the targeted graphs are hidden through estimating a density function of latent variables such that the pairwise distances in both the input space and the latent space are retained, and they have been successfully applied to various learning scenarios. However, previous work heuristically treated the multipliers in the dual as the graph weights, so the interpretation of this graph from a theoretical perspective is still missing. In this brief, we fill up this gap by presenting a detailed interpretation based on optimality conditions and their connections to neighborhood graphs. We further provide a systematic way to set up proper hyperparameters to prevent trivial graphs and achieve varied levels of sparsity. Three extensions are explored to leverage different measure functions, refine/reweigh an initial graph, and reduce computation cost for medium-sized graph. Extensive experiments on both synthetic and real datasets were conducted and experimental results verify our theoretical findings and the showcase of the studied graph in semisupervised learning provides competitive results to those of compared methods with their best graph.

Super Structure Fault-Tolerance Assessment of the Generalized Hypercube

Abstract Fault-tolerant performance of a network is the prerequisite and guarantee for the normal operation of a network, which is often characterized by connectivity. Let $H$ denote a connected subgraph of $G$ and $H^{*}$ denote the union of the set of all connected subgraphs of $H$ and the set of the trivial graph. Super $H$-connectivity (resp. super $H^{*}$-connectivity) satisfies the conditions of both super connectivity and $H$-structure connectivity (resp. $H$-substructure connectivity). These two kinds of new connectivity provide a new metric to measure the fault-tolerance of the network, that is, the super structure fault-tolerance. The generalized hypercube $G(m_{r}, m_{r-1},..., m_{1})$ is a universal topology of interconnection networks that contains other commonly used topologies and it has been applied in many data center networks because of its excellent qualities. In this paper, we research the super structure fault-tolerance of $G(m_{r}, m_{r-1},..., m_{1})$ by studying super $H$-connectivity $\kappa ^{\prime}(G|H)$ and super $H^{*}$-connectivity $\kappa ^{\prime}(G|H^{*})$ for $H\in \{K_{1,M},\ C_{3},\ C_{4},\ K_{4}\}$.

On Vietoris–Rips Complexes (with Scale 3) of Hypercube Graphs

On Vietoris–Rips Complexes (with Scale 3) of Hypercube Graphs

Flat GL(1|1)-connections and fatgraphs

We study the moduli space of flat GL(1|1)-connections on a punctured surface from the point of view of graph connections. To each fatgraph, a system of coordinates is assigned, which involves two bosonic and two fermionic variables per edge, subject to certain relations. In the case of trivalent graphs, we provide a closed explicit formula for the Whitehead moves. In addition, we discuss the invariant Poisson bracket.