Finite Vertex Research Articles

Elusive properties of infinite graphs

AbstractA graph property is said to be elusive (or evasive) if every algorithm testing this property by asking questions of the form “is there an edge between vertices and ?” requires, in the worst case, to ask about all pairs of vertices. The unsettled Aanderaa–Karp–Rosenberg conjecture is that every nontrivial monotone graph property is elusive for any finite vertex set. We show that the situation is completely different for infinite vertex sets: the monotone graph properties “every vertex has degree at least ” and “every connected component has size at least ,” where and are natural numbers, are not elusive for infinite vertex sets, but the monotone graph property “the graph contains a cycle” is elusive for arbitrary vertex set. On the other hand, we also prove that every algorithm testing some natural monotone graph properties, for example, “every vertex has degree at least ” or “connected” on the vertex set should check “lots of edges,” more precisely, all the edges of an infinite complete subgraph.

Ising percolation in the hyperbolic plane

We study infinite “+” or “−” clusters for an Ising model on an connected, transitive, non-amenable, planar, one-ended graph G with finite vertex degree. If the critical percolation probability pcsite for the independent identically distributed (IID). Bernoulli site percolation on G is less than 12, we find an explicit region for the coupling constant of the Ising model such that there are infinitely many infinite “+”-clusters and infinitely many infinite “−”-clusters, while the random cluster representation of the Ising model has no infinite 1-clusters. If pcsite>12, we obtain a lower bound for the critical probability in the random cluster representation of the Ising model in terms of pcsite. We also obtain an explicit region for the coupling constant when the XOR Ising model (the product of two IID Ising models) does not have a unique infinite contour a.s. and an explicit region for the coupling constant when the XOR Ising model has infinitely many infinite “+”-clusters and infinitely many infinite “−”-clusters.

Generalized Arithmetic Staircase Graphs and Their Total Edge Irregularity Strengths

Let Γ=(VΓ,EΓ) be a simple undirected graph with finite vertex set VΓ and edge set EΓ. A total n-labeling α:VΓ∪EΓ→{1,2,…,n} is called a total edge irregular labeling on Γ if for any two different edges xy and x′y′ in EΓ the numbers α(x)+α(xy)+α(y) and α(x′)+α(x′y′)+α(y′) are distinct. The smallest positive integer n such that Γ can be labeled by a total edge irregular labeling is called the total edge irregularity strength of the graph Γ. In this paper, we provide the total edge irregularity strength of some asymmetric graphs and some symmetric graphs, namely generalized arithmetic staircase graphs and generalized double-staircase graphs, as the generalized forms of some existing staircase graphs. Moreover, we give the construction of the corresponding total edge irregular labelings.

Countably determined ends and graphs

The directions of an infinite graph G are a tangle-like description of its ends: they are choice functions that choose a component of G−X for all finite vertex sets X⊆V(G) in a compatible manner.Although every direction is induced by a ray, there exist directions of graphs that are not uniquely determined by any countable subset of their choices. We characterise these directions and their countably determined counterparts in terms of star-like substructures or rays of the graph.Curiously, there exist graphs whose directions are all countably determined but which cannot be distinguished all at once by countably many choices.We structurally characterise the graphs whose directions can be distinguished all at once by countably many choices, and we structurally characterise the graphs whose directions cannot be distinguished in this manner. Our characterisations are phrased in terms of normal trees and tree-decompositions.Our four (sub)structural characterisations imply combinatorial characterisations of the four classes of infinite graphs that are defined by the first and second axiom of countability applied to their end spaces: the two classes of graphs whose end spaces are first countable or second countable, respectively, and the complements of these two classes.

Finiteness of spinfoam vertex amplitude with timelike polyhedra and the regularization of full amplitude

This work focuses on Conrady-Hnybida's 4-dimensional extended spinfoam model with timelike polyhedra, while we restrict all faces to be spacelike. Firstly, we prove the absolute convergence of the vertex amplitude with timelike polyhedra, when SU(1,1) boundary states are coherent states or the canonical basis, or their finite linear combinations. Secondly, based on the finite vertex amplitude and a proper prescription of the SU(1,1) intertwiner space, we construct the extended spinfoam amplitude on arbitrary cellular complex, taking into account the sum over SU(1,1) intertwiners of internal timelike polyhedra. We observe that the sum over SU(1,1) intertwiners is infinite for the internal timelike polyhedron that has at least 2 future-pointing and 2 past-pointing face-normals. In order to regularize the possible divergence from summing over SU(1,1) intertwiners, we develop a quantum cut-off scheme based on the eigenvalue of the ``shadow operator''. The spinfoam amplitude with timelike internal polyhedra (and spacelike faces) is finite, when 2 types of cut-offs are imposed: one is imposed on $j$ the eigenvalue of area operator, the other is imposed on the eigenvalue of shadow operator for every internal timelike polyhedron that has at least 2 future-pointing and 2 past-pointing face-normals.

Graphical construction of spatial Gibbs random graphs

We consider a random graph model on Zd that incorporates the interplay between the statistics of the graph and the underlying space where the vertices are located. Based on a graphical construction of the model as the invariant measure of a birth and death process, we prove the existence and uniqueness of a measure defined on graphs with vertices in Zd, which coincides with the limit along the measures over graphs with the finite vertex set. As a consequence, theoretical properties, such as exponential mixing of the infinite volume measure and central limit theorem for averages of a real-valued function of the graph, are obtained. Moreover, a perfect simulation algorithm based on the clan of ancestors is described in order to sample a finite window of the equilibrium measure defined on Zd.

Max Flows in Planar Graphs with Vertex Capacities

We consider the maximum flow problem in directed planar graphs with capacities on both vertices and arcs and with multiple sources and sinks. We present three algorithms when the capacities are integers. The first algorithm runs in O ( min { k 2 n , n log 3 n + kn }) time when all capacities are bounded by a constant, where n is the number of vertices in the graph, and k is the number of terminals. This algorithm is the first to solve the vertex-disjoint paths problem in linear time when k is fixed but larger than 2. The second algorithm runs in O ( k 5 Δ n polylog ( nU )) time, where each arc capacity and finite vertex capacity is bounded by U , and Δ is the maximum degree of the graph. Finally, when k = 3, we present an algorithm that runs in O ( n log n ) time; this algorithm works even when the capacities are arbitrary reals. Our algorithms improve on the fastest previously known algorithms when k and Δ are fixed and U is bounded by a polynomial in n . Prior to this result, the fastest algorithms ran in O ( n 4/3+ o (1) ) time for unit capacities; in the smallest of O ( n 3/2 log n log U ), Õ( n 10/7 U 1/7 ), O ( n 11/8+o(1) U 1/4 ), and O ( n 4/3 + o(1) U 1/3 ) time for integer capacities; and in O ( n 2 /log n ) time for real capacities, even when k = 3.

Duality theorems for stars and combs IV: Undominating stars

AbstractIn a series of four papers we determine structures whose existence is dual, in the sense of complementary, to the existence of stars or combs from the well‐known star—comb lemma for infinite graphs. Call a set of vertices in a graph tough in if only finitely many components of meet for every finite vertex set . In this fourth and final paper of the series, we structurally characterise the connected graphs in which a given vertex set is tough. Our characterisations are phrased in terms of tree‐decompositions, tangle‐distinguishing separators and tough subgraphs (a graph is tough if its vertex set is tough in ). From the perspective of stars and combs, we thereby find structures whose existence is complementary to the existence of so‐called undominating stars.

Locally interacting diffusions as Markov random fields on path space

We consider a countable system of interacting (possibly non-Markovian) stochastic differential equations driven by independent Brownian motions and indexed by the vertices of a locally finite graph G=(V,E). The drift of the process at each vertex is influenced by the states of that vertex and its neighbors, and the diffusion coefficient depends on the state of only that vertex. Such processes arise in a variety of applications including statistical physics, neuroscience, engineering and math finance. Under general conditions on the coefficients, we show that if the initial conditions form a second-order Markov random field on d-dimensional Euclidean space, then at any positive time, the collection of histories of the processes at different vertices forms a second-order Markov random field on path space. We also establish a bijection between (second-order) Gibbs measures on (Rd)V (with finite second moments) and a set of (second-order) Gibbs measures on path space, corresponding respectively to the initial law and the law of the solution to the stochastic differential equation. As a corollary, we establish a Gibbs uniqueness property that shows that for infinite graphs the joint distribution of the paths is completely determined by the initial condition and the specifications, namely the family of conditional distributions on finite vertex sets given the configuration on the complement. Along the way, we establish approximation and projection results for Markov random fields on locally finite graphs that may be of independent interest.

Constructing highly regular expanders from hyperbolic Coxeter groups

A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$ around $v$ is an $(a_1,\dots,a_{n-1})$-regular graph. Such a graph $X$ is said to be highly regular (HR) of level $n$ if $a_{n-1}\neq 0$. Chapman, Linial and Peled studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders globally and locally. They ask whether such HR-graphs of level 3 exist. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system $(W,S)$ and a subset $M$ of $S$, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope $\mathcal{P}_{W,M}$, which form an infinite family of expander graphs when $(W,S)$ is indefinite and $\mathcal{P}_{W,M}$ has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of $(W,S)$. The expansion stems from applying superapproximation to the congruence subgroups of the linear group $W$. This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked by Chapman, Linial and Peled.

A Finitary Structure Theorem for Vertex-Transitive Graphs of Polynomial Growth

We prove a quantitative, finitary version of Trofimov’s result that a connected, locally finite vertex-transitive graph Γ of polynomial growth admits a quotient with finite fibres on which the action of Aut(Γ) is virtually nilpotent with finite vertex stabilisers. We also present some applications. We show that a finite, connected vertex-transitive graph Γ of large diameter admits a quotient with fibres of small diameter on which the action of Aut(Γ) is virtually abelian with vertex stabilisers of bounded size. We also show that Γ has moderate growth in the sense of Diaconis and Saloff-Coste, which is known to imply that the mixing and relaxation times of the lazy random walk on Γ are quadratic in the diameter. These results extend results of Breuillard and the second author for finite Cayley graphs of large diameter. Finally, given a connected, locally finite vertex-transitive graph Γ exhibiting polynomial growth at a single, sufficiently large scale, we describe its growth at subsequent scales, extending a result of Tao and an earlier result of our own for Cayley graphs. In forthcoming work we will give further applications.

Complexity of signed total $k$-Roman domination problem in graphs

Let $G$ be a simple graph with finite vertex set $V(G)$ and $S = \{-1, 1, 2\}$. A signed total Roman $k$-dominating function (STRkDF) on a graph $G$ is a function $f:V(G)\to S$ such that (i) any vertex $y$ with $f(y) = -1$ is adjacent to at least one vertex $t$ with $f(t) = 2, $ (ii) $\sum_{t\in N(y)}f(t)\geq k$ holds for any vertex $y$. The $weight$ of an STRkDF $f$, denoted by $\omega(f)$, is $\sum_{y\in V(G)}f(y)$, and the minimum weight of an STRkDF is the <i>signed total Roman k-domination number</i>, $\gamma_{stR}^k(G), $ of $G$. In this article, we prove that the decision problem for the signed total Roman $k$-domination is NP-complete on bipartite and chordal graphs for $k\in\{1, 2\}$.

The α-normal labeling for generalized directed uniform hypergraphs

Let π=(ν1,…,νd) be an ordered partition of the integer m≥1, where ∑i=1dνi=m and νi∈Z+ for all i∈[d]. A π-directed m-uniform hypergraph G consists of a finite vertex set V(G) and a collection of edges E(G). Each edge is an ordered tuple, e=(S1(e),S2(e),…,Sd(e)), of disjoint subsets of vertices such that |Si|=νi, for all i∈[d]. For each edge e, |⋃i=1dSi(e)|=m. Moreover, ⋃e∈E(G)Si(e)=Si(G) and |Si(G)|=ni. Given p=(p1,p2,…,pd)∈(1,∞)d, the p-spectral radius of G is defined asλp(G):=max‖xi‖pi=1,i∈[d]⁡∑e∈E(G)∏i=1d∏v∈Si(e)xi,v, where xi=(xi,1,xi,2,…,xi,ni)∈Rni.In this paper, we develop the α-normal labeling method for calculating λp(G) and some related properties are given. Moreover, we give a new lower bound of the p-spectral radius of π-directed m-uniform hypergraphs for the case ∑i=1dνipi>1 by using the inverse Hölder's inequality.

The cubic graphs with finite cyclic vertex connectivity larger than girth

Cyclic (vertex and edge) connectivity is an important concept in graphs. While cyclic edge connectivity (cλ) has been studied for many years, the study at cyclic vertex connectivity (cκ) is still at the initial stage. And cκ seems to be more complicated than cλ. We have got a sufficient condition that ν(G)≥2g(k−1) for cκ≠∞. On the other hand, if ν(G)<2g(k−1), then we have cκ=∞, or cκ≤(k−2)g, or (k−2)g<cκ<∞. So characterizing all the k-regular graphs with (k−2)g<cκ<∞ is helpful to design an efficient algorithm for cκ. Hence, we characterize all 38 cubic graphs with g<cκ<∞ and prove that cκ=g+1.

Singular graphs with dihedral group action

Let Γ be a simple undirected graph on a finite vertex set and let A be its adjacency matrix. Then Γ is singular if A is singular. The problem of characterizing singular graphs is easy to state but very difficult to resolve in any generality. In this paper we investigate the singularity of graphs for which the dihedral group acts transitively on vertices as a group of automorphisms.

On indecomposable vertex algebras associated with vertex algebroids

Let A be a finite dimensional unital commutative associative algebra and let B be a finite dimensional vertex A-algebroid such that its Levi factor is isomorphic to sl2. Under suitable conditions, we construct an indecomposable non-simple N-graded vertex algebra VB‾ from the N-graded vertex algebra VB associated with the vertex A-algebroid B. We show that this indecomposable non-simple N-graded vertex algebra VB‾ is C2-cofinite and has only two irreducible modules.

Finite Vertex Bi-Primitive 2-Arc Transitive Graphs Admitting a Two-Dimensional Linear Group

A graph is said to be vertex bi-primitive, if it is a bipartite graph, and the setwise stabilizer of its automorphism group acts primitively on two bi-parts. In this paper, we not only classify vertex bi-primitive 2-arc transitive graphs admitting a two-dimensional linear group but also determine the automorphism groups and the number of non-isomorphic ones of such graphs.

Infinitely generated virtually free pro‐p groups and p‐adic representations

We prove the pro-$p$ version of the Karras, Pietrowski, Solitar, Cohen and Scott result stating that a virtually free group acts on a tree with finite vertex stabilizers. If a virtually free pro-$p$ group $G$ has finite centralizes of all non-trivial torsion elements more stronger statement is proved: $G$ embeds into a free pro-$p$ product of a free pro-$p$ group and finite $p$-group. The integral $p$-adic representation theory is used in the proof; it replaces the Stallings theory of ends in the pro-$p$ case.

Numerical integration and extrapolation for finite and UV-divergent 3-loop Feynman integrals

Automatic numerical integration attempts to alleviate the detailed application of solution methods to the problem. We use automatic multivariate integration with PARINT combined with numerical extrapolation for classes of 3-loop UV-divergent self-energy diagrams with massive internal lines, and finite or UV-divergent massless vertex diagrams. The extrapolation is performed with respect to the dimensional regularization parameter. It is shown that the PARINT software handles the integrations accurately and efficiently.

Cycles through all finite vertex sets in infinite graphs

A closed curve in the Freudenthal compactification |G| of an infinite locally finite graph G is called a Hamiltonian curve if it meets every vertex of G exactly once (and hence it meets every end at least once). We prove that |G| has a Hamiltonian curve if and only if every finite vertex set of G is contained in a cycle of G. We apply this to extend a number of results and conjectures on finite graphs to Hamiltonian curves in infinite locally finite graphs. For example, Barnette’s conjecture (that every finite planar cubic 3-connected bipartite graph is Hamiltonian) is equivalent to the statement that every one-ended planar cubic 3-connected bipartite graph has a Hamiltonian curve. It is also equivalent to the statement that every planar cubic 3-connected bipartite graph with a nowhere-zero 3-flow (with no restriction on the number of ends) has a Hamiltonian curve. However, there are 7-ended planar cubic 3-connected bipartite graphs that do not have a Hamiltonian curve.