All Questions
Tagged with mg.metric-geometry graph-theory
144 questions
0
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0
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25
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Is there a name for a spanner graph that only considers distance to a root node?
A $t$-spanner graph of a set of points $\{p_i\}$ in the plane is a graph $G = (V, E)$ such that for any pair of vertices $p_i, p_j \in V$, the shortest path distance $d_G(p_i, p_j)$ in $G$ is at most $...
- 4,049
4
votes
0
answers
66
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Convergence of graph geodesics to geodesics on metric spaces
Let $(X,d)$ be a compact length space metric space $\mathbb{X}_{\delta}$ be a $\delta$-packing on $X$ and, for every $k\in \mathbb{N}_+$, let $G_{k,\delta}=(\mathbb{X}_{\delta},\mathcal{E}_k,W_k)$ ...
- 364
3
votes
0
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147
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Understanding why $\frac{\phi^5}{2}$ solves this 3D optimization problem, where $\phi$ is the golden ratio
I would like to understand the deep meaning of a solution which arises from an optimization problem discussed in a paper of mine since it can be simply stated as $\frac{\phi^5}{2}$, where $\phi := \...
- 1,451
1
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0
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67
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Conjecture on the increasing efficiency of the shortest minimum-link polygonal chains covering any grids of the form $\{0,1,2\}^k$ as $k$ grows
From the well-known Nine dots problem, we know that we need a polygonal chain with at least $4$ edges to connect the $9$ points of the planar grid $G_{3,2}:=\{\{0, 1, 2\} \times \{0, 1, 2\}\} \subset \...
- 1,451
8
votes
1
answer
567
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Joining the $2^k$ points of $\{0,1\}^k$ with the shortest tree
Let $k$ be a given positive integer, and then consider the unit hypercube $\{0, 1\}^k \subset \mathbb{R}^k$ (i.e., a $k$-dimensional "cube" in the well-known Euclidean space).
We need to ...
- 1,451
6
votes
2
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404
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Estimating shortest paths in planar drawings of graphs
Consider a drawing (in $\mathbb{R}^2$) of a planar graph. (The drawing is given, contrarily to the common setup in graph theory where we are seeking to build a drawing with specific properties.)
For ...
- 307
10
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2
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255
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Is the face lattice of the cube a polytope graph?
The face lattice of a
convex polytope $P\subset\Bbb R^d$ is the partially ordered set whose elements are the faces of $P$ ordered by inclusion. We can turn it into a graph by considering its Hasse ...
- 13.6k
10
votes
3
answers
460
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Do triple-linked graphs exist?
Lets say that a finite simple graph $G$ is (intrinsically) fully triple-linked if for each embedding of $G$ into $\Bbb R^3$ we can find three disjoint cycles $C_1,C_2,C_3\subset G$ whose embeddings ...
- 13.6k
5
votes
0
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137
views
Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ are needed to uniquely determine all inner products
Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ need to be known to uniquely determine all inner products? I'll begin with the specific case I am ...
- 351
3
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0
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110
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Is every finite metric space representable in a pseudo-Euclidean space?
Let $X$ be a finite set with a (true) metric $d$ and $|X| = n$. Does there exist a set $Y$ of $n$ points in $R^n$ with a pseudo-Riemannian metric with signature $(n - k, k, 0)$ for some integer $k$ ...
- 31
1
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1
answer
213
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Qualitative values between two electrons in an atom or how to interpret these values?
This question is a little bit trying to understand physics through geometry of simplex:
Let $E_{i,j}$ be the ionization energy in times the number of hydrogen ionization energy for an element with ...
- 3,064
4
votes
2
answers
219
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Algorithm for grouping tetrahedra from Voronoi diagram
I have a set of 3D Voronoi generator points and their neighbouring points, which, when connected, should result in a Delaunay tetrahedralization. However, I'm having a hard time implementing this. My ...
- 142
1
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0
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125
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Do cycle graphs embed isometrically in spheres?
I recently came across, what seems to be a folklore. Namely, that cycle graphs embeds isometrically into spheres $S^n(r)$, for some $n\in \mathbb{N}_+$ and some $r>0$. However, I could not track ...
- 169
4
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2
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254
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Does the edge-graph of a centrally symmetric polytope determine which vertices are antipodal?
Given two origin symmetric convex polytopes $P_1$ and $P_2$ (that is $P_i=-P_i$) with the same edge-graph, but potentially of different dimensions and combinatorial types.
Let $\phi: G_{P_1}\to G_{P_2}...
- 13.6k
9
votes
0
answers
371
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Embedding a graph into Euclidean space
I want to find a map $v\mapsto \tilde v$ from the vertex set of a connected infinite graph $\Gamma$ to a Euclidean space that meets the following two conditions:
there is $\varepsilon>0$ such that ...
- 45k
4
votes
0
answers
132
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Can a polytopal graph be "centrally symmetric" in more than one way?
Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$.
The central symmetry of $P$ induces an involutory ...
- 13.6k
0
votes
0
answers
65
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Maximal number of times distance $1$ can occur among $n$ points in the plane [duplicate]
For $n\in\mathbb N$, let $f(n)$ be the maximal number of times distance $1$ can occur among $n$ points in the plane:
$$
f(n) = \max_{ \{ x_1,\ldots,x_n \} \subset \mathbb R^2} \# \big \{ i<j : \| ...
- 43.2k
6
votes
1
answer
257
views
Expected doubling constant of a random Erdős–Rényi graph
Consider the $G(n,p)$ random graph model where $n$ is a ``large'' positive integer and $p\in (0,1)$. We may equip every realized random graph $G$ with its shortest path distance, making it into a (...
- 5,405
1
vote
0
answers
48
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Inside-out dissection
In a recent problem in The College Math Journal (1230) a Heronian triangle is called to have an equivalent rectangle if there exists an integer sided rectangle with the same area and perimeter. For ...
- 111
3
votes
1
answer
135
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"Geodesic coherent" partition of a graph
Let $G=(V,E)$ be a finite undirected graph which we equip with its usual graph geodesic distance $d_G$ making $(G,d_G)$ into a metric space; let $1<\#V<\infty$. For a given $1<N< \#V$ ...
- 5,405
6
votes
1
answer
260
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Arbitrary-dimensional expanders?
Rephrasing expansion (slightly). Consider the following slightly tweaked version of the usual definition of a (spectral) expander graph.
(We write a weighted graph as $(V,\beta)$, where the weight $\...
- 20.2k
1
vote
0
answers
64
views
Angles between edges of a geometric graph and graph invariants
Are there any clever ways in which the angles between edges in a geometric graph are encoded in the graph spectrum, or another object associated with the graph?
I'm interested to see what else is ...
- 640
2
votes
0
answers
39
views
Estimating the largest radius making each ball in a finite metric space into a tree
Motivation:
Let $n$ be a positive integer and $(X,d)$ be an $n$-point metric space. Clearly, $(X,d)$ need not be a metric tree (e.g. take for example the discrete metric on $\{0,1,2\}$.
Conversely, ...
- 5,405
18
votes
2
answers
1k
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Are hyperbolic spaces actually better for embedding trees than Euclidean spaces?
There is a folklore in the empirical computer-science literature that, given a tree $(X,d)$, one can find a bi-Lipschitz embedding into a hyperbolic space $\mathbb{H}^n$ and that $n$ is "much ...
- 195
3
votes
1
answer
133
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Lattice-like structure with maximum spacing between vertices
I'll first describe my problem in layman's terms. I have a map with $m$ countries and I want to color each country with a different color (this has nothing to do with the 4-color theorem). How do I ...
- 3,098
3
votes
1
answer
180
views
When is a graph a $\operatorname{CAT}(\kappa)$ space?
Let $G:=(E,V,W)$ be a weighted graph and let $d_G$ be its graph metric, defined by on any two edges $e_1,e_2\in E$ by
$$
d_G(e_1,e_2)\triangleq
\inf_{\gamma}\, \sum_{v\in \gamma} W(v),\qquad\tag{0}\...
- 195
10
votes
3
answers
500
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Given the skeleton of an inscribed polytope. If I move the vertices so that no edge increases in length, can the circumradius still get larger?
Let $P\subset \Bbb R^n$ be an inscribed convex polytope, that is, all its vertices are on a common sphere of radius $r$.
Let $G$ be the edge-graph of $P$. For convenience, assume $V(G)=\{1,\dotsc,s\}$....
- 13.6k
15
votes
1
answer
1k
views
Ricci curvature : beyond heat-like flows
Let me give you some context first: just a few days ago I found some intriguing references to Ricci flows in the setting of directed graphs.
There are at least two versions of Ricci curvature in the ...
- 7,853
4
votes
1
answer
102
views
Shortcutting quasigeodesics
Let $\Gamma$ be a connected graph, let $\lambda \ge 1$ and $c \ge 0$ be some constants. Recall that a combinatorial path $p$ in $\Gamma$ is said to be $(\lambda,c)$-quasigeodesic if for every ...
- 3,181
3
votes
1
answer
143
views
Broken line that can go in specific directions: can it end up on its starting point?
Say you have a 2D broken line you move along, but only some directions are allowed (I give you the angles relative to the usual cartesian plane):
(Up-Left): $]\pi, \dfrac{\pi}{2}[$
(Down-Left): $]-\...
- 267
7
votes
1
answer
171
views
Metric TSP with integer edge cost
Given a metric TSP with integer edge cost upper-bounded by a constant $C_{\max}$, can we find an poly-time algorithm solving this TSP instance?
- 367
3
votes
2
answers
279
views
Construct by compactness (Pentagonal tiling – Rao paper)
In the (arXiv) paper, Exhaustive search of convex pentagons which tile the plane by Michael Rao, on page 4 under the proof of Lemma 2, it is said that:
"… We keep a connected component $H_d'$ of $...
- 133
0
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0
answers
81
views
Gromov–Hausdorff closure of non-positively curved graphs
Setup:
Let $\Gamma$ be the set of non-positively curved weighted connected graphs, with finitely many points, which are isometrically embedded in $\mathbb{R}^n$; for some $n\in \mathbb{N}$;$n\geq 2$. ...
- 5,405
10
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4
answers
1k
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An introductory text on expanders
I am looking for a book that covers expander graphs rigorously. Preferably a book aimed at beginners.
- 883
22
votes
2
answers
900
views
Is every 1-million-connected graph rigid in 3D?
It is an old result that every $6$-connected graph is rigid in $\mathbb{R}^2$:
Lovász, László, and Yechiam Yemini. "On generic rigidity in the plane." SIAM Journal on Algebraic Discrete ...
- 151k
7
votes
1
answer
283
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Are two quasi-isometric, isomorphic on large enough balls, transitive graphs isomorphic?
Take two transitive graphs $X,Y$ (potentially directed and edge-labelled, e.g. Cayley graphs).
Assume $X,Y$ are quasi-isometric with constant $K$, i.e. there exists a function $f:VX \to VY$ ($VX,\,VY$ ...
- 887
4
votes
1
answer
567
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Random graphs and Benjamini-Schramm convergence
I am looking for literature on the question whether a randomly chosen sequence of $k$-regular graphs converges in the Benjamini-Schramm sense to the universal covering with probability one.
There are ...
user130903
4
votes
3
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430
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How to show that random graphs cannot be embedded with short edges
For each (not necessarily planar) embedding of a graph in $\mathbb{R}^k$ one can calculate the ratio
$$\gamma = \frac{\textsf{mean Euclidean length of edges}}{\textsf{mean Euclidean distance between ...
- 9,720
2
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0
answers
115
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Sufficient coordinate-free condition for points being co-spheric
Question:
is there a theorem that guarantees that
$\mathcal{P}\subset\mathbb{E}^n$ is finite set of points in a Euclidean space and all radii of the $(n-1)$-spheres that are defined by the $n$-...
- 13.2k
0
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0
answers
70
views
Looking for a name for a generalization of geometry to graphs
I am pursuing generalizations of planar Euclidean geometry to complete symmetric and weighted graphs, the guiding principle being applicability to the TSP.
The operations and tests that are available ...
- 13.2k
2
votes
0
answers
129
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Decomposing a metric tree as a union of rooted (or "centered") trees
Suppose $G$ is a finite metric tree whose set of leaves is $A=\{v_1, \ldots, v_n\}$. Consider the function $G\to \mathbb R_+$ that assigns to a point $x$ the distance from $x$ to $A$, denoted $d(x, A)$...
- 10.9k
2
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0
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146
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What practically computable homotopy and/or (co)homology theories are known for finite (di)graphs, metric spaces, etc?
Of late I have taken to applying Dowker homology and the path homology theory of Grigor'yan et al. like a hammer to various relations and/or digraphs that have looked like nails. At the same time, I ...
3
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0
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134
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Two questions on counterexamples to Borsuk's conjecture and ball-packings
In 1933 Karol Borsuk conjectured the following
Can every bounded subset $E$ of $\mathbb{R}^d$ be partitioned into $(d+1)$ sets, each of which has a smaller diameter than $E$?
Whilst new to this ...
- 31
3
votes
1
answer
179
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When is a $k$-distance-transitive graph already distance-transitive?
Call a (finite and connected) graph $k$-distance-transitive if its symmetry group acts transitively on the pairs in each one of the sets
$$D_\delta:=\{(i,j)\in V\times V\mid \mathrm d(i,j)=\delta\},\...
- 13.6k
1
vote
0
answers
63
views
Gromov-Hausdorff distance between graphs with edges as part of the space versus not part of the space
Let $G_1$ and $G_2$ be finite simple graphs viewed as metric spaces in the natural way where the edges are not part of the space. Let $G_1'$ and $G_2'$ be copies of $G_1$ and $G_2$ resp. but with the ...
- 328
1
vote
1
answer
110
views
Distance pairs in labeled directed graph
Suppose we have a simple directed graph with $n$ nodes and $m$ edges, and we label each edge from $1$ to $m$ (with distinct labels). Define the weighted "length" of a directed path to be the maximum ...
- 130
11
votes
2
answers
669
views
Which curves and surfaces are realizable by linkages? references?
Ok, so I try to formulate rigorously the question in the title, for which I am asking for references. My definitions may be flawed, so feel free to adjust/correct them! I care about dimensions 2 and 3 ...
- 2,041
10
votes
2
answers
496
views
Graph metric approximating Euclidean metric
I've been reading Wolfram's recent articles about graph/mesh/grid structures as an analogy for physical space, and it seems to me that there will be a problem getting the notion of distance to work ...
- 119
3
votes
1
answer
203
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Voronoi diagram on (weighted) graphs
Suppose I have a graph $G$ (possibly with weights on edges), and I have a subset $S$ of $k$ vertices $s_1, \dotsc, s_k$. I want to solve the post office problem: that is, I want to partition the ...
- 96.4k
7
votes
0
answers
102
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Median spaces as retracts of hypercubes
It is known (See e.g. here, Theorem 2.1) that median graphs are retracts of hypercubes.
Question: Is it also known that median metric spaces are retract of some $l¹$ product of unit intervals?
By ...
- 887