Andrés Aranda, David Bradley-Williams, Eng Keat Hng, Jan Hubička, Miltiadis Karamanlis, Michael Kompatscher, Matěj Konečný, Micheal Pawliuk
We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions.
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Completing Partial Packings of Bipartite Graphs
给定一个二部图$H$和一个整数$n$，设$f(n;H)$是最小整数，使得$H$在$n$顶点上的任何边不相交副本集，可以扩展为至多$n+f(n;H)$顶点上的$H$ -设计.我们建立了$f(n;H)$作为$n \rightarrow \infty$的增长的严格界限。特别地，我们证明了Füredi和Lehel的\cite{FuLe}猜想：$f(n;H) = o(n)$ .这解决了一个长期悬而未决的问题。
The $c_2$-invariant is an arithmetic graph invariant useful for understanding Feynman periods. Brown and Schnetz conjectured that the $c_2$-invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of the $c_2$-invariant in the $p=2$ case, extending previous work of one of us. The methods are combinatorial and enumerative involving counting certain partitions of the edges of the graph.
Completing partial $k$-star designs
本文研究部分k-星设计的补全问题，对所有参数k和n确定了n阶不可补全部分k-星设计所需星的最少数量。
Completing the Boundary Case of the Mahmoodian-Mirzakhani Conjecture and 117 New Computational 5-Cycle Decompositions of Complete Tripartite Graphs
本文证明了Mahmoodian-Mirzakhani猜想的边界情形，构造了117个完全三部图的5-圈分解，解决了大量此前未解决的相关三元组的分解问题。
We investigate Ramsey expansions, the coherent extension property for partial isometries (EPPA), and the existence of a stationary independence relation for all classes of metrically homogeneous graphs from Cherlin's catalogue. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and coherent EPPA. Our results are a contribution to Nešetřil's classification programme of Ramsey classes and c…
Metric dimension is a graph parameter motivated by problems in robot navigation, drug design, and image processing. In this paper, we answer several open extremal problems on metric dimension and pattern avoidance in graphs from (Geneson, Metric dimension and pattern avoidance, Discrete Appl. Math. 284, 2020, 1-7). Specifically, we construct a new family of graphs that allows us to determine the maximum possible degree of a graph of metric dimension at most $k$, the maximum possible degeneracy of a graph of metric dimension at most $k$, the maximum possible chromatic number of a graph of metri…
Maximum Common Subelement Metrics and its Applications to Graphs
本文刻画了一个称为最大公共子元（MCS）模型的数学模型，并证明了该模型上四种不同度量的存在性。我们推广了以前在文献中提出的图上的度量，并通过基于（1）子图，（2）诱导子图和（3）子图的扩展概念的图上的MCS模型的三个不同示例来识别新的度量。后一个示例可以用于对具有复杂标签的图进行建模（例如，其标签是其他图的图），并因此导出它们上的度量。此外，我们还使用（3）来表明，当度量时，图编辑距离与相应MCS模型中的最大公共子元素有关。
Distance Graphs of Metric Spaces with Rosenbloom - Tsfasman metric
Rosenbloom和Tsfasman提出了一个新的度量（RT度量），它是Hamming度量的推广。本文研究了空间$Z_q^n$和空间$S_n$在Rosenbloom -Tsfasman度量下的距离图.我们还描述了这些图的顶点度、分支度和色数。
In this paper we determine the metric dimension of Andrasfai graphs and some related graphs.
On the fractional metric dimension of graphs
在[S. Arumugam，V.Mathew和J.Shen，On Fractional Metric Dimension of Graphs，预印本]，Arumugam等人研究了两个图的笛卡儿积的分数度量维数，提出了四个公开问题.本文研究了点传递图的分数度量维数，特别是点传递距离正则图的分数度量维数可以用它的相交数表示.作为应用，我们分别计算了Hamming图和Johnson图的分数度量维数。此外，我们给出了任意图的度量维数与分数度量维数的一个不等式，并确定了当不等式成立时的所有图。最后，我们建立了图的笛卡儿积的分数度量维数的界。从而彻底解决了这四个开放性问题。
In this paper, we study the metric dimension of Cayley graphs. Specially, we present a complete characterization of Cayley graphs on Abelian groups whose metric dimension is two.
In this paper extremal values of the difference between several graph invariants related to the metric dimension are studied: mixed metric dimension, edge metric dimension and strong metric dimension. These non-trivial extremal values are computed over all connected graphs of given order. To obtain such extremal values several techniques are developed. They use functions related to metric dimension graph invariants to obtain lower and/or upper bounds on these extremal values and exact computations when restricting to some specific families of graphs.
The metric (resp. edge metric) dimension of a simple connected graph $G$, denoted by dim$(G)$ (resp. edim$(G)$), is the cardinality of a smallest vertex subset $S\subseteq V(G)$ for which every two distinct vertices (resp. edges) in $G$ have distinct distances to a vertex of $S$. It is an interesting topic to discuss the relation between dim$(G)$ and edim$(G)$ for some class of graphs $G$. In this paper, we settle two open problems on this topic for a widely studied class of graphs, called unicyclic graphs. Specifically, we introduce four classes of subgraphs to characterize the structure of a…
We consider the localization number and metric dimension of certain graphs of diameter $2$, focusing on families of Kneser graphs and graphs without 4-cycles. For the Kneser graphs with diameter $2$, we find upper and lower bounds for the localization number and metric dimension, and in many cases these parameters differ only by an additive constant. Our results on the metric dimension of Kneser graphs improve on earlier ones, yielding exact values in infinitely many cases. We determine bounds on the localization number and metric dimension of Moore graphs of diameter $2$ and polarity graphs.
For a given graph $G$, the metric and edge metric dimensions of $G$, $\dim(G)$ and ${\rm edim}(G)$, are the cardinalities of the smallest possible subsets of vertices in $V(G)$ such that they uniquely identify the vertices and the edges of $G$, respectively, by means of distances. It is already known that metric and edge metric dimensions are not in general comparable. Infinite families of graphs with pendant vertices in which the edge metric dimension is smaller than the metric dimension are already known. In this article, we construct a 2-connected graph $G$ such that $\dim(G)=a$ and ${\rm e…
Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance vectors to such a set. In this work, we settle three open problems on (edge) metric dimension of graphs. Specifically, we show that for every $r,t\ge 2$ with $r\ne t$, there is $n_0$, such that for every $n\ge n_0$ there exists a graph $G$ of order $n$ with metric dimension $r$ and edge metric dimension $t$, which among other consequences, shows the existence of …
Etzion et al. introduced metrics on $\mathbb{F}_2^n$ based on directed graphs on $n$ vertices and developed some basic coding theory on directed graph metric spaces. In this paper, we consider the problem of classifying directed graphs which admit the extended Hamming codes to be a perfect code. We first consider weighted poset metrics as a natural generalization of poset metrics and investigate interrelation between weighted poset metrics and directed graph based metrics. In the next, we classify weighted posets on a set with eight elements and directed graphs on eight vertices which admit th…
On the (adjacency) metric dimension of corona and strong product graphs and their local variants: combinatorial and computational results
度量维数是一个研究得很好的图参数。最近，引入了邻接度量维数和局部度量维数。我们结合这些变种，并引入本地邻接度量维度。我们证明了一个$n$阶图和一些非平凡图$H$的冠积的（局部）度量维数等于$n$乘以$H$的（局部）邻接度量维数.这种强关系也使我们能够推断计算（本地）度量维度的计算硬度结果，根据硬度结果（本地）邻接度量维度，我们也提供。我们还研究了图的强积的组合性质，并强调不同类型的孪生子在确定特别是图的邻接度量维数中所起的作用。
On the Metric Dimension of Infinite Graphs
一个顶点集$S$ \emph{resolves}称为图$G$，如果每个顶点都由它到$S$中顶点的距离向量唯一确定。图$G$的\emph{metric dimension}是分解集的最小基数。本文研究了顶点度有限的无限图的度量维数。给出了这类图具有有限度量维数的必要条件，刻画了具有有限度量维数的无限树。建立了有限图和无限图的笛卡儿积的度量维数的一些结果，并给出了几类图的笛卡儿积的度量维数。
We investigate how the metric dimension of infinite graphs change when we add edges to the graph. Our two main results: (1) there exists a growing sequence of graphs (under the subgraph relation, but without adding vertices) for which the metric dimension changes between finite and infinite infinitely many times; (2) finite changes in the edge set can not change the metric dimension from finite to infinite or vice versa.
We characterize the strong metric dimension of the power graph of a finite group. As applications, we compute the strong metric dimension of the power graph of a cyclic group, an abelian group, a dihedral group or a generalized quaternion group.
On the metric dimension of bilinear forms graphs
一个图的度量维数是一个集合中顶点的最小数目，它的性质是从任何顶点到集合中的顶点的距离列表唯一地标识该顶点。Bailey和Meagher得到了Grassmann图的度量维数的一个上界。本文给出了双线性型图的度量维数的一个上界。
Given a graph $G = (V,E)$, a set $S \subset V$ is called a $k$-\emph{metric generator} for $G$ if any pair of different vertices of $G$ is distinguished by at least $k$ elements of $S$. A graph is $k$-\emph{metric dimensional} if $k$ is the largest integer such that there exists a $k$-metric generator for $G$. This paper studies some bounds on the number $k$ for which a graph is $k$-metric dimensional.
Computing the local metric dimension of a graph from the local metric dimension of primary subgraphs
对于顶点的有序子集$W = \{w_1, w_2,\dots w_k\}$和连通图$G$中的顶点$u$，$u$关于$W$的表示是有序$k$元组$ r(u|W)=(d(v,w_1), d(v,w_2),\dots,$ $d(v,w_k))$，其中$d(x,y)$表示顶点$x$和$y$之间的距离。如果$G$的每两个相邻顶点具有不同的表示，则集合$W$是$G$的局部度量生成器。最小局部度量生成元称为$G$的\emph{local metric basis}，其基数为G的\emph{local metric dimension}。我们表明，计算的局部度量维数的图与割顶点减少到所谓的主子图的局部度量维数的计算。主要结果被应用到具体的建设，包括花束的图，根积图，冠积图，块图和链的图。
The Diametral Metric Dimension of Generalized Corona Graph
本研究定义并求解广义冠状图的直径度量维数，发现其取决于主图直径与分支图度量维数，为基础设施规划提供理论支撑。
Metric dimension of some distance-regular graphs
一个图的分解集是一个顶点的集合，它具有这样的性质，即从任何顶点到集合中的顶点的距离列表唯一地标识该顶点。本文分别构造了Johnson图、双Odd图、双Grassmann图和扭曲Grassmann图的分解集，并得到了这些图的度量维数的上界.
Many concrete problems are formulated in terms of a finite set of points in $R^n$ which, via the ambient Euclidean metric, becomes a finite metric space. To obtain information from such a space, it is often useful to associate a graph to it, and do mathematics on the graph, rather than on the space. Connected graphs become finite metric spaces (rather, their set of vertices) via the path-metric. We consider different types of connected graphs that can be associated to a metric space. In turn, the metric spaces obtained from these graphs recover, in some cases, identically the initial space, wh…
A signed graph $Σ$ is a pair $(G,σ)$, where $G=(V,E)$ is the underlying graph in which each edge is assigned $+1$ or $-1$ by the signature function $σ:E\rightarrow\{-1,+1\}$. In this paper, we extend the extensively applied concepts of metric dimension and resolving sets for unsigned graphs to signed graphs. We analyze the metric dimension of some well known classes of signed graphs including a special case of signed trees. Among other things, we establish that the metric dimension of a signed graph is invariant under negation.
Topics concerning metric dimension related invariants in graphs are nowadays intensively studied. This compendium of combinatorial and computational results on this topic is an attempt of surveying those contributions that are of the highest interest for the research community dealing with several variants of metric dimension in graphs.
The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order $n$ whose metric subgraphs are all paths if and only if $n\ge 13$ and the smallest size of such a graph of order $13$ is $22;$ (2) there exists a graph of order $n$ whose metric subgraphs are all cycles if and only if $n\ge 15,$ and there are exactly three such graphs of order $15;$ (3) for every integer $k\ge 3,$ we determine the possible orders for the existence of a graph whose metric subgraphs are all conne…
Nonlocal metric dimension ${\rm dim}_{\rm n\ell}(G)$ of a graph $G$ is introduced as the cardinality of a smallest nonlocal resolving set, that is, a set of vertices which resolves each pair of non-adjacent vertices of $G$. Graphs $G$ with ${\rm dim}_{\rm n\ell}(G) = 1$ or with ${\rm dim}_{\rm n\ell}(G) = n(G)-2$ are characterized. The nonlocal metric dimension is determined for block graphs, for corona products, and for wheels. Two upper bounds on the nonlocal metric dimension are proved. An embedding of an arbitrary graph into a supergraph with a small nonlocal metric dimension and small dia…
A resolving set for a graph $Γ$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The metric dimension of $Γ$ is the smallest size of a resolving set for $Γ$. Much attention has been paid to the metric dimension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimension of primitive distance-regular graphs in terms of their parameters. We show how the metric dimension of an imprimitive distance-regular graph can be related to that of its halved and f…
We prove a tropical analogue of the theorem of Hurwitz: a leafless metric graph of genus $g \ge 2$ has at most $12$ automorphisms when $g = 2$; $2^g g!$ automorphisms when $g \ge 3$. These inequalities are optimal; for each genus, we give all metric graphs which have the maximum numbers of automorphisms. The proof is written in terms of graph theory.
The study of combinatorial games is intimately tied to the study of graphs, as any game can be realized as a directed graph in which players take turns traversing the edges until reaching a sink. However, there have heretofore been few efforts towards analyzing game graphs using graph theoretic metrics and techniques. A set $S$ of vertices in a graph $G$ resolves $G$ if every vertex in $G$ is uniquely determined by the vector of its distances from the vertices in $S$. A metric basis of $G$ is a smallest resolving set and the metric dimension is the cardinality of a metric basis. In this articl…
Borgs, Chayes, Cohn and Holden (2016+) recently extended the definition of graphons from probability spaces to arbitrary $σ$-finite measure spaces, in order to study limits of sparse graphs. They also extended the definition of the cut metric, and proved various results on the resulting metric space. We continue this line of research and give various further results on graphons and the cut metric in this general setting, extending known results for the standard case of graphons on probability spaces. In particular, we characterize pairs of equivalent graphons, and we give new results on comple…
Metric dimension for random graphs
图$G$的度量维数是$G$的顶点集的子集$S$中的顶点的最小数目，使得所有其他顶点由它们到$S$中的顶点的距离唯一地确定。本文研究了随机图$G(n,p)$的度量维数，给出了概率$p=p(n)$在很宽范围内的度量维数.
The metric dimension is quite a well-studied graph parameter. Recently, the adjacency dimension and the local metric dimension have been introduced and studied. In this paper, we give a general formula for the local metric dimension of the lexicographic product $G \circ \mathcal{H}$ of a connected graph $G$ of order $n$ and a family $\mathcal{H}$ composed by $n$ graphs. We show that the local metric dimension of $G \circ \mathcal{H}$ can be expressed in terms of the true twin equivalence classes of $G$ and the local adjacency dimension of the graphs in $\mathcal{H}$.
On the strong metric dimension of the complement of the zero-divisor graph of a lattice
本文计算布尔格膨胀的零因子图补图的强度量维数，并由此推导出全图、极大图、理想交集图、约化环零因子图补图及向量空间分量图的强度量维数。
Exploring Metric and Strong Metric Dimensions in Inclusion Ideal Graphs of Commutative Rings
本文研究交换环包含理想图$In(R)$，讨论其度量维数，刻画解析图结构，并据此计算强度量维数。
In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G)[E(G) is called the mixed metric dimension of G. In this paper we first establish the exact value of the mixed metric dimension of a unicycic graph G which is derived from the structure of G. We further consider graphs G with edge disjoint cycles in which a unicyclic restriction Gi is introduced for each cycle Ci: Applying the result for unicyclic graph to each Gi then yields the exact value of the mixed metric dimension of such a graph G. The obtained formulas for the exact value of …
Computing the metric dimension of a graph from primary subgraphs
设$G$是一个连通图.给定有序集$W = \{w_1, w_2,\dots w_k\}\subseteq V(G)$和顶点$u\in V(G)$，$u$相对于$W$的表示是有序$k$元组$(d(u,w_1), d(u,w_2),\dots,$ $d(u,w_k))$，其中$d(u,w_i)$表示$u$和$w_i$之间的距离.如果$G$的每两个不同的顶点具有不同的表示，则集合$W$是$G$的度量生成器。最小基数度量生成器称为$G$的\emph{metric basis}，其基数称为G的\emph{metric dimension}。图的度量维数是一个NP难问题。本文给出了割点图的度量维数的封闭公式。主要结果被应用到具体的建设，包括根产品图，冠产品图，块图和链的图。
The concept of metric dimension has applications in a variety of fields, such as chemistry, robotic navigation, and combinatorial optimization. We show bounds for graphs with $n$ vertices and metric dimension $β$. For Hamiltonian outerplanar graphs, we have $β\leq \left\lceil\frac{n}2\right\rceil$; for outerplanar graphs in general, we have $β\leq \left\lfloor\frac{2n}{3}\right\rfloor$; for maximal planar graphs, we have $β\leq \left\lfloor\frac{3n}{4}\right\rfloor$. We also show that bipyramids have a metric dimension of $\left\lfloor\frac{2n}{5}\right\rfloor + 1$. It is conjectured that the …
Let $G=(V,E)$ be a connected graph. A vertex $w\in V$ distinguishes two elements (vertices or edges) $x,y\in E\cup V$ if $d_G(w,x)\ne d_G(w,y)$. A set $S$ of vertices in a connected graph $G$ is a mixed metric generator for $G$ if every two elements (vertices or edges) of $G$ are distinguished by some vertex of $S$. The smallest cardinality of a mixed metric generator for $G$ is called the mixed metric dimension and is denoted by $\mathrm{mdim}(G)$. In this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial…
Construction of tropical morphisms from tropical modifications of nonhyperelliptic genus 3 metric graphs with tree gonality 3 to metric trees
该研究针对树亚次数为3的非超椭圆亏格3度量图，构造了其对应的热带修改、度量树以及次数为3的热带态射，完善了热带曲线亚次数相关理论。
In two recent papers by Veitch and Roy and by Borgs, Chayes, Cohn, and Holden, a new class of sparse random graph processes based on the concept of graphexes over $σ$-finite measure spaces has been introduced. In this paper, we introduce a metric for graphexes that generalizes the cut metric for the graphons of the dense theory of graph convergence. We show that a sequence of graphexes converges in this metric if and only if the sequence of graph processes generated by the graphexes converges in distribution. In the course of the proof, we establish a regularity lemma and determine which sets …
We determine the metric dimension of the annihilating-ideal graph of a local finite commutative principal ring and a finite commutative principal ring with two maximal ideals. We also find the bounds for the metric dimension of the annihilating-ideal graph of an arbitrary finite commutative principal ring.
Eigenvalue bracketing for discrete and metric graphs
我们开发的拉普拉斯算子的特征值估计离散和度量图使用不同类型的边界条件在顶点的度量图。通过一个明确的对应关系的等边度量和离散图的频谱（也在“例外”的值的度量图对应的狄利克雷谱），我们进行这些估计从度量图拉普拉斯到离散的情况。我们将结果应用到覆盖图和目前的例子，覆盖图拉普拉斯算子有谱间隙。
Spectral analysis of metric graphs and related spaces
本文的目的是在离散图和流形的谱几何的指导下，给出度量图的谱理论的一个概述。我们提出了度量图和自然拉普拉斯算子的基本概念，并明确允许无限图。受度量图上拉普拉斯算子一般形式的启发，我们定义了一类新的组合拉普拉斯算子。有了这个广义离散拉普拉斯算子，就有可能把离散图和度量图的谱理论联系起来。此外，我们描述了度量图与流形的连接。最后，我们评论Cheeger的不等式和跟踪公式的度量和离散（广义）拉普拉斯算子。
Metric graphs are often introduced based on combinatorics, upon "associating" each edge of a graph with an interval; or else, casually "gluing" a collection of intervals at their endpoints in a network-like fashion. Here we propose an abstract, self-contained definition of metric graph. Being mostly topological, it doesn't require any knowledge from graph theory and already determines uniquely several concepts that are commonly and unnecessarily \textit{defined} in the literature. Nevertheless, many ideas mentioned here are folklore in the quantum graph community: we discuss them for later ref…
In this paper, the leading term of the asymptotics of the number of possible final positions of a random walk on a directed Hamiltonian metric graph is found. Consideration of such dynamical systems could be motivated by problems of propagation of narrow wave packets on metric graphs.
A certain class of directed metric graphs is considered. Asymptotics for a number of possible endpoints of a random walk at large times is found.
We prove that metric graph with the minimal growth of the number of possible endpoints of a random walk is the union of several linear paths coming out of the same vertex
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