Cubic Graphs Research Articles

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.

Feature aggregation graph convolutional network based on imaging genetic data for diagnosis and pathogeny identification of Alzheimer's disease.

The roles of brain regions activities and gene expressions in the development of Alzheimer's disease (AD) remain unclear. Existing imaging genetic studies usually has the problem of inefficiency and inadequate fusion of data. This study proposes a novel deep learning method to efficiently capture the development pattern of AD. First, we model the interaction between brain regions and genes as node-to-node feature aggregation in a brain region-gene network. Second, we propose a feature aggregation graph convolutional network (FAGCN) to transmit and update the node feature. Compared with the trivial graph convolutional procedure, we replace the input from the adjacency matrix with a weight matrix based on correlation analysis and consider common neighbor similarity to discover broader associations of nodes. Finally, we use a full-gradient saliency graph mechanism to score and extract the pathogenetic brain regions and risk genes. According to the results, FAGCN achieved the best performance among both traditional and cutting-edge methods and extracted AD-related brain regions and genes, providing theoretical and methodological support for the research of related diseases.

Conical $SL(3)$ foams

In the unoriented SL(3) foam theory, singular vertices are generic singularities of two-dimensional complexes. Singular vertices have neighborhoods homeomorphic to cones over the one-skeleton of the tetrahedron, viewed as a trivalent graph on the two-sphere. In this paper we consider foams with singular vertices with neighborhoods homeomorphic to cones over more general planar trivalent graphs. These graphs are subject to suitable conditions on their Kempe equivalence Tait coloring classes and include the dodecahedron graph. In this modification of the original homology theory it is straightforward to show that modules associated to the dodecahedron graph are free of rank 60, which is still an open problem for the original unoriented SL(3) foam theory.

Approximating sparse graphs: The random overlapping communities model

AbstractHow can we approximate sparse graphs and sequences of sparse graphs (with unbounded average degree)? We consider convergence in the first moments of the graph spectrum (equivalent to the numbers of closed ‐walks) appropriately normalized. We introduce a simple random graph model that captures the limiting spectra of many sequences of interest, including the sequence of hypercube graphs. The random overlapping communities (ROC) model is specified by a distribution on pairs , . A graph on vertices with average degree is generated by repeatedly picking pairs from the distribution, adding an Erdős–Rényi random graph of edge density on a subset of vertices chosen by including each vertex with probability , and repeating this process so that the expected degree is . We also show that ROC graphs exhibit an inverse relationship between degree and clustering coefficient, a characteristic of many real‐world networks.

Edge Isoperimetric Inequalities for Powers of the Hypercube

For positive integers $n$ and $r$, we let $Q_n^r$ denote the $r$th power of the $n$-dimensional discrete hypercube graph, i.e., the graph with vertex-set $\{0,1\}^n,$ where two 0-1 vectors are joined if they are Hamming distance at most $r$ apart. We study edge isoperimetric inequalities for this graph. Harper, Bernstein, Lindsey and Hart proved a best-possible edge isoperimetric inequality for this graph in the case $r=1$. For each $r \geq 2$, we obtain an edge isoperimetric inequality for $Q_n^r$; our inequality is tight up to a constant factor depending only upon $r$. Our techniques also yield an edge isoperimetric inequality for the `Kleitman-West graph' (the graph whose vertices are all the $k$-element subsets of $\{1,2,\ldots,n\}$, where two $k$-element sets have an edge between them if they have symmetric difference of size two); this inequality is sharp up to a factor of $2+o(1)$ for sets of size $\binom{n -s}{k-s}$, where $k=o(n)$ and $s \in \mathbb{N}$.

Tangled piecewise-linear embeddings of trivalent graphs.

A method is described for generating and exploring tangled piecewise-linear embeddings of trivalent graphs under the constraints of point-group symmetry. It is shown that the possible vertex-transitive tangles are either graphs of vertex-transitive polyhedra or bipartite vertex-transitive nonplanar graphs. One tangle is found for 6 vertices, three for 8 vertices (tangled cubes), seven for 10 vertices, and 21 for 12 vertices. Also described are four isogonal embeddings of pairs of cubes and 12 triplets of tangled cubes (16 and 24 vertices, respectively). Vertex 2-transitive embeddings are obtained for tangled trivalent graphs with 6 vertices (two found) and 8 vertices (45 found). Symmetrical tangles of the 10-vertex Petersen graph and the 20-vertex Desargues graph are also described. Extensions to periodic tangles are indicated. These are all interesting and viable targets for molecular synthesis.

An effective multiple-image encryption algorithm based on 3D cube and hyperchaotic map

In accordance with the principle that the cube graph can be obtained by superimposing planes, a multiple-image encryption scheme based on hyperchaotic map and 3D cube is designed in this paper. Multiple images are segmented by columns, and a regular cube is obtained by stacking multiple fixed-size planes. Each stacked plane is considered as a stacked plane of the cube along the z-axis direction. Firstly, each tangent plane is rotated by different angles in the counterclockwise direction and then DNA encoded, followed by face swap, row swap, and column swap in turn. Then, the DNA-encoded image cube is subjected to DNA addition with the chaotic cube. Finally, two DNA sequences are randomly selected for DNA mutation. The mutated cube is decoded into a digital cube by DNA and the multiple-image encryption algorithm is completed. Theoretically, the cube used in the encryption process can be infinitely large, which means that the number and the size of 2D images that can be encrypted can be unrestricted. Experimental simulations and performance tests prove that the proposed image encryption scheme is effective and secure.

Partition function of the Levin-Wen model

Using a description of the Levin-Wen model excitations in terms of Wilson lines, we compute the degeneracy of the energy levels for any input anyon theory and for any trivalent graph embedded on any (orientable) compact surface. This result allows one to obtain the finite-size and finite-temperature partition function and to show that there are no thermal phase transitions.

Eccentric Harmonic Index for the Cartesian Product of Graphs

Suppose ρ is a simple graph, then its eccentric harmonic index is defined as the sum of the terms 2/ea + eb for the edges vavb, where ea is the eccentricity of the ath vertex of the graph ρ. We symbolize the eccentric harmonic index (EHI) as He = He(ρ). In this article, we determine He for the Cartesian product (CP) of particularly chosen graphs. Lower bounds for He of the CP of the two graphs are established. The formulas of EHI for the Hamming and Hypercube graphs are obtained. These obtained formulas can be used in QSAR and QSPR studies to get a better understanding of their applications in mathematical chemistry.

The sensitivity conjecture, induced subgraphs of cubes, and Clifford algebras

We give another version of Huang’s proof that an induced subgraph of the {n} -dimensional cube graph containing over half the vertices has maximal degree at least {\sqrt{n}} , which implies the Sensitivity Conjecture. This argument uses Clifford algebras of positive definite signature in a natural way. We also prove a weighted version of the result.

Signed domination number of hypercubes

The [Formula: see text]-cube graph or hypercube [Formula: see text] is the graph whose vertex set is the set of all [Formula: see text]-dimensional Boolean vectors, two vertices being joined if and only if they differ in exactly one co-ordinate. The purpose of the paper is to investigate the signed domination number of this hypercube graphs. In this paper, signed domination number [Formula: see text]-cube graph for odd [Formula: see text] is found and the lower and upper bounds of hypercube for even [Formula: see text] are found.

On Vietoris–Rips complexes of hypercube graphs

We describe the homotopy types of Vietoris–Rips complexes of hypercube graphs at small scale parameters. In more detail, let \(Q_n\) be the vertex set of the hypercube graph with \(2^n\) vertices, equipped with the shortest path metric. Equivalently, \(Q_n\) is the set of all binary strings of length n, equipped with the Hamming distance. The Vietoris–Rips complex of \(Q_n\) at scale parameter zero is \(2^n\) points, and the Vietoris–Rips complex of \(Q_n\) at scale parameter one is the hypercube graph, which is homotopy equivalent to a wedge sum of circles. We show that the Vietoris–Rips complex of \(Q_n\) at scale parameter two is homotopy equivalent to a wedge sum of 3-spheres, and furthermore we provide a formula for the number of 3-spheres. Many questions about the Vietoris–Rips complexes of \(Q_n\) at larger scale parameters remain open.

Tropically planar graphs

We study tropically planar graphs, which are the graphs that appear in smooth tropical plane curves. We develop necessary conditions for graphs to be tropically planar, and compute the number of tropically planar graphs up to genus 7. We provide non-trivial upper and lower bounds on the number of tropically planar graphs, and prove that asymptotically \(0\%\) of connected trivalent planar graphs are tropically planar.

A Combinatorial Approach to the Computation of the Fractional Edge Dimension of Graphs

E. Yi recently introduced the fractional edge dimension of graphs. It has many applications in different areas of computer science such as in sensor networking, intelligent systems, optimization, and robot navigation. In this paper, the fractional edge dimension of vertex and edge transitive graphs is calculated. The class of hypercube graph Qn with an odd number of vertices n is discussed. We propose the combinatorial criterion for the calculation of the fractional edge dimension of a graph, and hence applied it to calculate the fractional edge dimension of the friendship graph Fk and the class of circulant graph Cn(1,2).

Codes, Cubes, and Graphical Designs

Graphical designs are an extension of spherical designs to functions on graphs. We connect linear codes to graphical designs on cube graphs, and show that the Hamming code in particular is a highly effective graphical design. We show that even in highly structured graphs, graphical designs are distinct from the related concepts of extremal designs, maximum stable sets in distance graphs, and t-designs on association schemes.

Quantum tunneling and quantum walks as algorithmic resources to solve hard K-SAT instances

We present a new quantum heuristic algorithm aimed at finding satisfying assignments for hard K-SAT instances using a continuous time quantum walk that explicitly exploits the properties of quantum tunneling. Our algorithm uses a Hamiltonian H_A(F) which is specifically constructed to solve a K-SAT instance F. The heuristic algorithm aims at iteratively reducing the Hamming distance between an evolving state {|{psi _j}rangle } and a state that represents a satisfying assignment for F. Each iteration consists on the evolution of {|{psi _j}rangle } (where j is the iteration number) under e^{-iH_At}, a measurement that collapses the superposition, a check to see if the post-measurement state satisfies F and in the case it does not, an update to H_A for the next iteration. Operator H_A describes a continuous time quantum walk over a hypercube graph with potential barriers that makes an evolving state to scatter and mostly follow the shortest tunneling paths with the smaller potentials that lead to a state {|{s}rangle } that represents a satisfying assignment for F. The potential barriers in the Hamiltonian H_A are constructed through a process that does not require any previous knowledge on the satisfying assignments for the instance F. Due to the topology of H_A each iteration is expected to reduce the Hamming distance between each post measurement state and a state {|{s}rangle }. If the state {|{s}rangle } is not measured after n iterations (the number n of logical variables in the instance F being solved), the algorithm is restarted. Periodic measurements and quantum tunneling also give the possibility of getting out of local minima. Our numerical simulations show a success rate of 0.66 on measuring {|{s}rangle } on the first run of the algorithm (i.e., without restarting after n iterations) on thousands of 3-SAT instances of 4, 6, and 10 variables with unique satisfying assignments.

Super fault-tolerance assessment of locally twisted cubes based on the structure connectivity

The connectivity of a connected graph G is the minimum cardinality over all vertex-cuts, which can be determined using Menger's theorem (1927). Then G is super connected if its minimum vertex-cut is always composed of a vertex's neighborhood. Two generalized extensions to this classic notion of connectivity include the T-structure connectivity κ(G;T) and the T-substructure connectivity κs(G;T), for which T is the given structure isomorphic to a connected subgraph of G. Let T⁎ denote the union of the set of all connected subgraphs of T and the set of the trivial graph. In this article, a connected graph G is called super T-connected if the minimum degree of G−F is zero for each minimum T-cut F of G; analogously, G is super T⁎-connected if the minimum degree of G−F is zero for each minimum T⁎-cut F of G. Considering the n-dimensional locally twisted cube LTQn with n≥3, we first establish both κ(LTQn;T) and κs(LTQn;T) and then determine whether LTQn is super T-connected and super T⁎-connected, where T∈{K1,1,K1,2,K1,3,C4}∪{Pk|4≤k≤n}.

G-family polynomials

We introduce two notions of quandle polynomials for [Formula: see text]-families of quandles: the quandle polynomial of the associated quandle and a [Formula: see text]-family polynomial with coefficients in the group ring of [Formula: see text]. As an application we define image subquandle polynomial enhancements of the [Formula: see text]-family counting invariant for trivalent spatial graphs and handlebody-links. We provide examples to show that the new enhancements are proper.

A diagrammatic presentation and its characterization of non-split compact surfaces in the 3-sphere

We give a presentation for a non-split compact surface embedded in the 3-sphere [Formula: see text] by using diagrams of spatial trivalent graphs equipped with signs and we define Reidemeister moves for such signed diagrams. We show that two diagrams of embedded surfaces are related by Reidemeister moves if and only if the surfaces represented by the diagrams are ambient isotopic in [Formula: see text].

Prism graphs in tropical plane curves

Any smooth tropical plane curve contains a distinguished trivalent graph called its skeleton. In 2020 Morrison and Tewari proved that the so-called big face graphs cannot be the skeleta of tropical curves for genus $12$ and greater. In this paper we answer an open question they posed to extend their result to the prism graphs, proving that they are the skeleton of a smooth tropical plane curve precisely when the genus is at most $11$. Our main tool is a classification of lattice polygons with two points than can simultaneously view all others, without having any one point that can observe all others.