Finite Graph Research Articles

A NOTE ON THE STRONG ERDŐS–HAJNAL PROPERTY FOR GRAPHS WITH BOUNDED VC-MINIMAL COMPLEXITY

Abstract Inspired by Adler’s idea on VC minimal theories [1], we introduce VC-minimal complexity. We show that for any $N\in \mathbb {N}^{>0}$ , there is $k_N>0$ such that for any finite bipartite graph $(X,Y;E)$ with VC-minimal complexity $< N$ , there exist $X'\subseteq X$ , $Y'\subseteq Y$ with $|X'|\geq k_N |X|$ , $|Y'|\geq k_N |Y|$ such that $X'\times Y' \subseteq E$ or $X'\times Y'\cap E=\emptyset $ .

BOREL LINE GRAPHS

Abstract We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the nine finite graphs from the classical result of Beineke together with a 10th infinite graph associated with the equivalence relation $\mathbb {E}_0$ on the Cantor space. As a corollary, we prove a partial converse to the Feldman–Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers.

Dispersive effects for the Schrödinger equation on finite metric graphs with infinite ends

We study the free Schrödinger equation on finite metric graphs with infinite ends. We give sufficient conditions to obtain the L1(R)→L∞(R) time decay rate at least t−1/2. These conditions allow certain metric graphs with circles and/or with commensurable lengths of the bounded edges. Further we study the dynamics of the probability flow between the bounded sub-graph and the unbounded ends.

Some Generator Subgraphs of the Square of a Cycle

Graphs considered in this paper are finite simple graphs, which has no loops and multiple edges. Let $G= (V(G),E(G))$ be a graph with $E(G) = \{e_{1}, e_{2},\ldots, e_{m}\},$ for some positive integer $m.$ The \textit{edge space} of $G,$ denoted by $\mathscr{E}(G),$ is a vector space over the field $\zn_{2}.$ The elements of $\mathscr{E}(G)$ are all the subsets of $E(G)$. Vector addition is defined as $X+Y = X ~\Delta~ Y,$ the symmetric difference of sets $X$ and $Y,$ for $X,Y \in \mathscr{E}(G).$ Scalar multiplication is defined as $1\cdot X =X$ and $0 \cdot X = \emptyset$ for $X \in \mathscr{E}(G).$ Let $H$ be a subgraph of $G.$ The \textit{uniform set of $H$} with respect to $G,$ denoted by $E_{H}(G),$ is the set of all elements of $\mathscr E(G)$ that induces a subgraph isomorphic to $H.$ The subspace of $\mathscr E(G)$ generated by $E_{H}(G)$ shall be denoted by $\mathscr E_{H}(G).$ If $E_H(G)$ is a generating set, that is $\mathscr E_{H}(G)= \mathscr E(G),$ then $H$ is called a \textit{generator subgraph} of $G.$ This paper determines some generator subgraphs of the square of a cycle. Moreover, this paper established sufficient conditions for the generator subgraphs of the square of a cycle.

SPANNING TREES IN $\mathbb {Z}$ -COVERS OF A FINITE GRAPH AND MAHLER MEASURES

Abstract Using the special value at $u=1$ of Artin–Ihara L-functions, we associate to every $\mathbb {Z}$ -cover of a finite connected graph a polynomial, which we call the Ihara polynomial. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce–Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specialising to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and I-graphs (including the generalised Petersen graphs). We also express the p-adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the p-adic Mahler measure of the Ihara polynomial. When applied to a particular $\mathbb {Z}$ -cover, our result gives us back Lengyel’s calculation of the p-adic valuations of Fibonacci numbers.

Weak Randomness in Graphons and Theons

ABSTRACTCall a hereditary family of graphs strongly persistent if there exists a graphon such that in all subgraphons of , is precisely the class of finite graphs that have positive density in . Our first result is a complete characterization of the hereditary families of graphs that are strongly persistent as precisely those that are closed under substitutions. We call graphons with the self‐similarity property above weakly random. A hereditary family is said to have the weakly random Erdős–Hajnal property () if every graphon that is a limit of graphs in has a weakly random subgraphon. Among families of graphs that are closed under substitutions, we completely characterize the families that belong to as those with “few” prime graphs. We also extend some of the results above to structures in finite relational languages by using the theory of theons.

Towards Nash‐Williams orientation conjecture for infinite graphs

AbstractIn 1960 Nash‐Williams proved that an edge‐connectivity of is sufficient for a finite graph to have a ‐arc‐connected orientation. He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on the auxiliary lifting graph, proved that ‐edge‐connected infinite graphs admit a ‐arc‐connected orientation. Here we improve this result for the class of one‐ended locally finite graphs and show that an edge‐connectivity of is enough in that case. Crucial to this improvement are results presented in a separate paper, by the same author of this paper, on the key concept of the lifting graph, extending results by Ok, Richter, and Thomassen.

LGTCN: A Spatial–Temporal Traffic Flow Prediction Model Based on Local–Global Feature Fusion Temporal Convolutional Network

High-precision traffic flow prediction facilitates intelligent traffic control and refined management decisions. Previous research has built a variety of exquisite models with good prediction results. However, they ignore the reality that traffic flows can propagate backwards on road networks when modeling spatial relationships, as well as associations between distant nodes. In addition, more effective model components for modeling temporal relationships remain to be developed. To address the above challenges, we propose a local–global features fusion temporal convolutional network (LGTCN) for spatio-temporal traffic flow prediction, which incorporates a bidirectional graph convolutional network, probabilistic sparse self-attention, and a multichannel temporal convolutional network. To extract the bidirectional propagation relationship of traffic flow on the road network, we improve the traditional graph convolutional network so that information can be propagated in multiple directions. In addition, in spatial global dimensions, we propose probabilistic sparse self-attention to effectively perceive global data correlations and reduce the computational complexity caused by the finite perspective graph. Furthermore, we develop a multichannel temporal convolutional network. It not only retains the temporal learning capability of temporal convolutional networks, but also corresponds each channel to a node, and it realizes the interaction of node features through output interoperation. Extensive experiments on four open access benchmark traffic flow datasets demonstrate the effectiveness of our model.

A general multi-scale description of metastable adaptive motion across fitness valleys

We consider a stochastic individual-based model of adaptive dynamics on a finite trait graph G=(V,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=(V,E)$$\\end{document}. The evolution is driven by a linear birth rate, a density dependent logistic death rate and the possibility of mutations along the directed edges in E. We study the limit of small mutation rates for a simultaneously diverging population size. Closing the gap between Bovier et al. (Ann Appl Probab 29(6):3541–358, 2019) and Coquille et al. (Electron J Probab 26:1–37, 2021) we give a precise description of transitions between evolutionary stable conditions (ESC), where multiple mutations are needed to cross a valley in the fitness landscape. The system shows a metastable behaviour on several divergent time scales, corresponding to the widths of these fitness valleys. We develop the framework of a meta graph that is constituted of ESCs and possible metastable transitions between them. This allows for a concise description of the multi-scale jump chain arising from concatenating several jumps. Finally, for each of the various time scales, we prove the convergence of the population process to a Markov jump process visiting only ESCs of sufficiently high stability.

Pro-[formula omitted] RAAGs

Let C be a class of finite groups closed under taking subgroups, quotients, and extensions with abelian kernel. The right-angled Artin pro-C group GΓ (pro-C RAAG for short) is the pro-C completion of the right-angled Artin group G(Γ) associated with the finite simplicial graph Γ.In the first part, we describe structural properties of pro-C RAAGs. Among others, we describe the centraliser of an element and show that pro-C RAAGs satisfy the Tits' alternative, that standard subgroups are isolated, and that 2-generated pro-p subgroups of pro-C RAAGs are either free pro-p or free abelian pro-p.In the second part, we characterise splittings of pro-C RAAGs in terms of the defining graph. More precisely, we prove that a pro-C RAAG GΓ splits as a non-trivial direct product if and only if Γ is a join and it splits over an abelian pro-C group if and only if a connected component of Γ is a complete graph or it has a complete disconnecting subgraph. We then use this characterisation to describe an abelian JSJ decomposition of a pro-C RAAG, in the sense of Guirardel and Levitt [GL17].

Results on super edge magic deficiency of somewell-known classes of finite graphs

A graph Ω(Λ, Γ) is considered super edge magic if there exists a bijective function ϕ:Λ(Ω)∪Γ(Ω)→ {1, 2, 3, …,|Λ(Ω)|+|Γ(Ω)|} such that ϕ(τ1)+(τ1τ2)+ ϕ(τ2) is a constant for every edge τ1τ2 ∈ Γ(Ω), and ϕ(Λ(Ω))= {1, 2, 3, …,|Λ(Ω)|}. Furthermore, the super edge magic deficiency of a graph Ω, denoted as μs (Ω), is either the minimum non-negative integer η such that Ω∪ηK1 is a super edge magic graph or +∞ if such an integer η does not exist. In this paper, we investigate the super edge magic deficiency of certain families of graphs.

Neighbor Locating Coloring on Graphs: Three Products

Let G be a simple finite graph. A k -coloring of G is a partition π = { S 1 , ⋯ , S k } of V ( G ) so that each S i is an independent set and any vertex in S i takes color i . A k -coloring π = { S 1 , ⋯ , S k } of V ( G ) is a neighbor locating coloring if for any two vertices u , v ∈ S i , there is a color class S j for which one of them has a neighbor in S j and the other does not. The minimum k with this property is said to be the neighbor locating chromatic number of G , denoted by χ N L ( G ) . We initiate the study of the neighbor locating coloring of graphs resulting from three types of product of two graphs. We investigate the neighbor locating chromatic number of Cartesian, lexicographic, and corona products of two graphs. Finally, we untangle the neighbor locating chromatic number of any of the aforementioned three products of cycles, paths, and complete graphs.

Ubiquity of locally finite graphs with extensive tree-decompositions

Ubiquity of locally finite graphs with extensive tree-decompositions

On the exponential growth of the number of finite asymmetric Hamiltonian graphs

Combinatorial analysis of various algebraic systems and their properties is an urgent problem of modern algebra. In a number of works, the growth function of algebraic systems has been studied, which, for a given natural n, indicates the number of n-element finite algebraic systems of a given type. In many works, the actual problem of studying various transformations of algebraic systems has also been investigated. In particular, it was investigated how "rich" or, conversely, "poor" the considered algebraic systems are with transformations of one kind or another. The paper studies asymmetric Hamiltonian graphs with only an identical automorphism. The article proves that the exponential growth function of asymmetric Hamiltonian graphs is. Комбинаторный анализ различных алгебраических систем и их свойств является актуальной проблемой современной алгебры. В ряде работ изучалась функция роста алгебраических систем, которая для заданного натурального n указывает число n-элементных конечных алгебраических систем заданного вида. Во многих работах исследовалась также актуальная проблема изучения различных преобразований алгебраических систем. В частности, исследовалось, насколько «богаты» или, наоборот, «бедны» рассматриваемые алгебраические системы преобразованиями того или иного вида. В статье изучаются асимметричные гамильтоновы графы, обладающие только тождественным автоморфизмом. В статье доказано, что является экспоненциальной функция роста асимметричных гамильтоновых графов.

Note on intrinsic metrics on graphs

AbstractWe study the set of intrinsic metrics on a given graph. This is a convex compact set and it carries a natural order. We investigate existence of largest elements with respect to this order. We show that the only locally finite graphs which admit a largest intrinsic metric are certain finite star graphs. In particular, all infinite locally finite graphs do not admit a largest intrinsic metric. For infinite graphs which are not locally finite the set of intrinsic metrics may be trivial as we show by an example. Moreover, we give a characterization for the existence of intrinsic metrics with finite balls for weakly spherically symmetric graphs.

Supercritical percolation on finite transitive graphs I: Uniqueness of the giant component

Supercritical percolation on finite transitive graphs I: Uniqueness of the giant component

The heat kernel on a finite graph in different Time-scales

The heat kernel on a finite graph in different Time-scales

Gradient flows of generalized relative entropy and functional inequalities on graphs

We derive the discrete forms of the porous medium equation and fast diffusion equation with drift terms on a finite graph as gradient flows of the m-relative entropy, with the global existence of solutions. We show the Łojasiewicz inequality to prove the convergence of solutions and some important functional inequalities. More specifically, the Łojasiewicz inequality holds near the stationary solution with exponent 12, which leads to the exactly exponential convergence rate with a finite trajectory length. It can also be applied to show the Talagrand-type inequality and the log-Sobolev-type inequality directly.

Automorphisms of Graph Products of Groups and Acylindrical Hyperbolicity

This article is dedicated to the study of the acylindrical hyperbolicity of automorphism groups of graph products of groups. Our main result is that, if Γ \Gamma is a finite graph which contains at least two vertices and is not a join and if G \mathcal {G} is a collection of finitely generated irreducible groups, then either Γ G \Gamma \mathcal {G} is infinite dihedral or A u t ( Γ G ) \mathrm {Aut}(\Gamma \mathcal {G}) is acylindrically hyperbolic. This theorem is new even for right-angled Artin groups and right-angled Coxeter groups. Various consequences are deduced from this statement and from the techniques used to prove it. For instance, we show that the automorphism groups of most graph products verify vastness properties such as being SQ-universal; we show that many automorphism groups of graph products do not satisfy Kazhdan’s property (T); we solve the isomorphism problem between graph products in some cases; and we show that a graph product of coarse median groups, as defined by Bowditch, is coarse median itself.

The matroid of a graphing

Graphings serve as limit objects for bounded-degree graphs. We define the “cycle matroid” of a graphing as a submodular setfunction, with values in [0,1], which generalizes (up to normalization) the cycle matroid of finite graphs. We prove that for a Benjamini–Schramm convergent sequence of graphs, the total rank, normalized by the number of nodes, converges to the total rank of the limit graphing.