Cubic Graphs Research Articles

Graph Comparison Meets Alexandrov

Graph comparison is a certain type of condition on the metric space encoded by a finite graph. We show that each nontrivial graph comparison implies one of Alexandrov’s comparisons. The proof gives a complete description of graphs with trivial graph comparisons.

Linear arboricity of degenerate graphs

AbstractA linear forest is a union of vertex‐disjoint paths, and the linear arboricity of a graph , denoted by , is the minimum number of linear forests into which the edge set of can be partitioned. Clearly, for a graph with maximum degree . On the other hand, the Linear Arboricity Conjecture due to Akiyama, Exoo, and Harary from 1981 asserts that for every graph . This conjecture has been verified for planar graphs and graphs whose maximum degree is at most , or is equal to or . Given a positive integer , a graph is ‐degenerate if it can be reduced to a trivial graph by successive removal of vertices with a degree at most . We prove that for any ‐degenerate graph provided .

Families of diffeomorphisms and concordances detected by trivalent graphs

AbstractWe study families of diffeomorphisms detected by trivalent graphs via the Kontsevich classes. We specify some recent results and constructions of the second named author to show that those non‐trivial elements in homotopy groups are lifted to homotopy groups of the moduli space of ‐cobordisms . As a geometrical application, we show that those elements in for are also lifted to the rational homotopy groups of the moduli space of positive scalar curvature metrics. Moreover, we show that the same elements come from the homotopy groups of moduli space of concordances of positive scalar curvature metrics on with fixed‐round metric on the boundary .

Quantum walk processes in quantum devices

Simulation and programming of current quantum computers as Noisy Intermediate-Scale Quantum (NISQ) devices represent a hot topic at the border of current physical and information sciences. The quantum walk process represents a basic subroutine in many quantum algorithms and plays an important role in studying physical phenomena. Simulating quantum walk processes is computationally challenging for classical processors. With an increasing improvement in qubits fidelity and qubits number in a single register, there is a potential to improve quantum walks simulations substantially. However, efficient ways to simulate quantum walks in qubit registers still have to be explored. Here, we explore the relationship between quantum walk on graphs and quantum circuits. Firstly, we discuss ways to obtain graphs provided quantum circuit. We then explore techniques to represent quantum walk on a graph as a quantum circuit. Specifically, we study hypercube graphs and arbitrary graphs. Our approach to studying the relationship between graphs and quantum circuits paves way for the efficient implementation of quantum walks algorithms on quantum computers.

The (Strong) Rainbow Connection Number of Join Of Ladder and Trivial Graph

Let G = (V,E) be a nontrivial, finite, and connected graph. A function c from E to {1,2,...,k},k ∈ N, can be considered as a rainbow k-coloring if every two vertices x and y in G has an x- y path. Therefore, no two path's edges receive the same color; this condition is called a “rainbow path”. The smallest positive integer k, designated by rc(G), is the G rainbow connection number. Thus, G has a rainbow k-coloring. Meanwhile, the c function is considered as a strong rainbow k-coloring within the condition for every two vertices x and y in G have an x - y rainbow path whose length is the distance between x and y. The smallest positive integer k, such as G, has a strong rainbow k-coloring; such a condition is called a strong rainbow connection number of G, denoted by src(G). In this research, the rainbow connection number and strong rainbow connection number are determined from the graph resulting from the join operation between the ladder graph and the trivial graph, denoted by rc(L_n∨K_1) and src(L_n∨K_1) respectively. So, rc (L_n∨K_1 )= src (L_n∨K_1 )=2,"for" 3≤n≤4 and rc (L_n∨K_1 )=3, while src(L_n∨K_1 )=⌈n/2⌉,"for" n≥5.

Constellation Design With Hypercube Graphs

Constellation Design With Hypercube Graphs

Counting geodesics of given commutator length

Abstract Let $\Sigma $ be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in $\Sigma $ having at most length L and which can be written as the product of g commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in $\Sigma $ . In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem.

The total vertex irregularity strength of symmetric cubic graphs of the Foster's Census

<p>Foster (1932) performed a mathematical census for all connected symmetric cubic (trivalent) graphs of order <em>n</em> with <em>n </em>≤ 512. This census then was continued by Conder et al. (2006) and they obtained the complete list of all connected symmetric cubic graphs with order <em>n</em> ≤ 768. In this paper, we determine the total vertex irregularity strength of such graphs obtained by Foster. As a result, all the values of the total vertex irregularity strengths of the symmetric cubic graphs of order <em>n</em> from Foster census strengthen the conjecture stated by Nurdin, Baskoro, Gaos & Salman (2010), namely ⌈(<em>n</em>+3)/4⌉.</p>

Iterative Construction of the Minimum Independent Dominating Sets in Hypercube Graphs

Iterative Construction of the Minimum Independent Dominating Sets in Hypercube Graphs

HyperHMM: Efficient inference of evolutionary and progressive dynamics on hypercubic transition graphs.

The evolution of bacterial drug resistance and other features in biology, the progression of cancer and other diseases, and a wide range of broader questions can often be viewed as the sequential stochastic acquisition of binary traits (for example, genetic changes, symptoms, or characters). Using potentially noisy or incomplete data to learn the sequences by which such traits are acquired is a problem of general interest. The problem is complicated for large numbers of traits which may, individually or synergistically, influence the probability of further acquisitions both positively and negatively. Hypercubic inference approaches, based on hidden Markov models on a hypercubic transition network, address these complications, but previous Bayesian instances can consume substantial time for converged results, limiting their practical use. Here we introduce HyperHMM, an adapted Baum-Welch (expectation maximisation) algorithm for hypercubic inference with resampling to quantify uncertainty, and show that it allows orders-of-magnitude faster inference while making few practical sacrifices compared to previous hypercubic inference approaches. We show that HyperHMM allows any combination of traits to exert arbitrary positive or negative influence on the acquisition of other traits, relaxing a common limitation of only independent trait influences. We apply this approach to synthetic and biological datasets and discuss its more general application in learning evolutionary and progressive pathways. Code for inference and visualisation, and data for example cases, is freely available at https://github.com/StochasticBiology/hypercube-hmm. Supplementary Information for this article is available online.

On metric dimensions of hypercubes

In this note we show two unexpected results concerning the metric, the edge metric and the mixed metric dimensions of hypercube graphs. First, we show that the metric and the edge metric dimensions of Qd differ by at most one for every integer d. In particular, if d is odd, then the metric and the edge metric dimensions of Qd are equal. Second, we prove that the metric and the mixed metric dimensions of the hypercube Qd are equal for every d ≥ 3. We conclude the paper by conjecturing that all these three types of metric dimensions of Qd are equal when d is large enough.

Some examples of quantum graphs

We summarize different approaches to the theory of quantum graphs and provide several ways to construct concrete examples. First, we classify all undirected quantum graphs on the quantum space $$M_2$$ . Secondly, we apply the theory of 2-cocycle deformations to Cayley graphs of abelian groups. This defines a twisting procedure that produces a quantum graph, which is quantum isomorphic to the original one. For instance, we define the anticommutative hypercube graphs. Thirdly, we construct an example of a quantum graph, which is not quantum isomorphic to any classical graph.

Minimal generating sets of moves for diagrams of isotopic knots and spatial trivalent graphs

Polyak proved that all oriented versions of Reidemeister moves for knot and link diagrams can be generated by a set of just four oriented Reidemeister moves, and that no fewer than four oriented Reidemeister moves generate them all. We refer to a set containing four oriented Reidemeister moves that collectively generate all of the other oriented Reidemeister moves as a minimal generating set. Polyak also proved that a certain set containing two Reidemeister moves of type 1, one move of type 2 and one move of type 3 form a minimal generating set for all oriented Reidemeister moves. We expand upon Polyak’s work by providing an additional eleven minimal, 4-element, generating sets of oriented Reidemeister moves, and we prove that these 12 sets represent all possible minimal generating sets of oriented Reidemeister moves. We also consider the Reidemeister-type moves that relate oriented spatial trivalent graph diagrams with trivalent vertices that are sources and sinks and prove that a minimal generating set of oriented Reidemeister-type moves for spatial trivalent graph diagrams contains 10 moves.

Scalar curvature operator for models of loop quantum gravity on a cubical graph

In this article we introduce a new operator representing the three-dimensional scalar curvature in loop quantum gravity. Our construction does not apply to the entire kinematical Hilbert space of loop quantum gravity; instead, the operator is defined on the Hilbert space of a fixed cubical graph. The starting point of our work is to write the spatial Ricci scalar classically as a function of the densitized triad and its SU(2)-covariant derivatives. We pass from the classical expression to a quantum operator through a regularization procedure, in which covariant derivatives of the triad are discretized as finite differences of gauge covariant flux variables on the cubical lattice provided by the graph. While more work is needed in order to extend our construction to encompass states based on all possible graphs, the operator presented here can be applied in models such as quantum-reduced loop gravity and effective dynamics, which are derived from the kinematical framework of full loop quantum gravity, and are formulated in terms of states defined on cubical graphs.

Arc-transitive trivalent Cayley graphs

A combinatorial graph Γ is symmetric, or arc-transitive, if its automorphism group acts transitively on the arcs of Γ, and s-arc-transitive (resp. s-arc-regular) if its automorphism group acts transitively (resp. regularly) on the set of s-arcs of Γ, which are the walks of length s in Γ in which any three consecutive vertices are distinct. It was shown by Tutte (1947, 1959) that every finite symmetric trivalent graph is s-arc-regular for some s≤5. Djoković and Miller (1980) took this further by showing that there are seven types of arc-transitive group action on finite trivalent graphs, characterised by the stabilisers of a vertex and an edge. The latter classification was refined by Conder and Nedela (2009), in terms of what types of arc-transitive subgroup can occur in the automorphism group of Γ.In this paper we address the question of when a finite trivalent Cayley graph is arc-transitive, by determining when a connected finite arc-transitive trivalent graph is a Cayley graph. We show that in five of the 17 Conder-Nedela classes, there is no Cayley graph, while in two others, every graph is a Cayley graph. In eight of the remaining ten classes, we give necessary conditions on the order of the graph for it to be Cayley; there is no such condition in the other two. Also we show that in each of those last ten classes, there are infinitely many Cayley graphs and infinitely many non-Cayley graphs.

Construction of continuum from a discrete surface by its iterated subdivisions

Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we introduce a subdivision method by applying the Goldberg-Coxeter subdivision and discuss the convergence of a sequence of discrete surfaces defined inductively by the subdivision. We also study the limit set as the continuum geometric object associated with the given discrete surface.

The continuous-time quantum walk on some graphs based on the view of quantum probability

In this paper, continuous-time quantum walk is discussed based on the view of quantum probability, i.e. the quantum decomposition of the adjacency matrix A of graph. Regard adjacency matrix A as Hamiltonian which is a real symmetric matrix with elements 0 or 1, so we regard [Formula: see text] as an unbiased evolution operator, which is related to the calculation of probability amplitude. Combining the quantum decomposition and spectral distribution [Formula: see text] of adjacency matrix A, we calculate the probability amplitude reaching each stratum in continuous-time quantum walk on complete bipartite graphs, finite two-dimensional lattices, binary tree, [Formula: see text]-ary tree and [Formula: see text]-fold star power [Formula: see text]. Of course, this method is also suitable for studying some other graphs, such as growing graphs, hypercube graphs and so on, in addition, the applicability of this method is also explained.

Decomposition of hypercube graphs into paths and cycles of length four

By a -decomposition of a graph G, we mean a partition of the edge set of G into p paths of length 4 and q cycles of length 4. In this paper, we give conditions for a -decomposition of the n-dimensional hypercube graph Qn for all possible values of p and q. In particular, we find the necessary and sufficient conditions for such decomposition in Qn when n is even, and n is odd,

The cycle-structure connectivity of crossed cubes

Crossed cube is a kind of network structure as a basis for distributed memory parallel computer architecture. Connectivity plays an important role in measuring the fault-tolerance of an interconnection network. As a generalization, Lin et al. [6] introduced the concept of structure connectivity in 2016. For connected graphs G and T, the T-structure connectivity κ(G;T) of G is the minimum cardinality of a set of subgraphs F of G that each member is isomorphic to T so that G−F is a disconnected or a trivial graph. In this paper, we study cycle-structure connectivity of n-dimensional crossed cubes CQn. We show that κ(CQn;C2k−1)=⌈nk−1⌉ for 4⩽k⩽n and κ(CQn;C2k)=⌊nk⌋+1 for even k with 4⩽k⩽n. Furthermore, we also obtain that κ(CQn;C6)=⌊n4⌋+⌈n−3⌊n4⌋2⌉ for n⩾4.

On factor-invariant graphs with two cycles

We classify trivalent vertex-transitive graphs whose edge sets have a partition into a 2-factor composed of two cycles and a 1-factor that is invariant under the action of the automorphism group.