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Results tagged with graph-theory
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user 2672
Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
40
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7
answers
9k
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Spectral graph theory: Interpretability of eigenvalues and -vectors
I thought "Wow!" when I learned that the eigenvector of the adjacency matrix of a cycle graph $C_n$ corresponding to the second largest eigenvalue gives the coordinates of the vertices when equally di …
- 105
7
votes
1
answer
1k
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Can every finite graph be represented by an arithmetic sequence of natural numbers?
(This is a follow-up to my previous questions Natural models of graphs?.)
Erdös in The Representation of a Graph by Set Intersections (1966) states:
Theorem. Let $G$ be an arbitrary
graph. Then there …
- 1,446
5
votes
2
answers
168
views
Abstract characterization of polygonizations
Consider a polygonization of the plane by convex polygons of a given minimal size that meet edge-to-edge and vertex-to-vertex.
What's the “official” name of such a polygonization?
Such polygonization …
- 1,446
9
votes
3
answers
13k
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Number of trees with n nodes and m leaves
Even searching for " 'number of trees' leaves " didn't reveal what I am looking for: an approach for calculating the (approximate) number of trees with exactly n nodes and m leaves. Any hints from MO? …
- 4,713
3
votes
1
answer
214
views
Pairs of paths with the same source and target
Commutative diagrams usually express path equivalences in a category and thus involve pairs of paths in a category with the same source and target.
General diagrams - in categories resp. category theo …
- 4,713
5
votes
4
answers
2k
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The distance distribution of graphs
The degree distribution of a graph is of main importance, especially for large graphs, and namely random graphs. Its expected value and its higher moments tell a lot about a graph – but of course not …
- 1,444
22
votes
4
answers
3k
views
Can you determine whether a graph is the 1-skeleton of a polytope?
How do I test whether a given undirected graph is the 1-skeleton of a polytope?
How can I tell the dimension of a given 1-skeleton?
- 4,713
1
vote
0
answers
71
views
Another betweenness centrality measure: neighbourhood centrality
Among the many centrality measures that I have heard of, I miss the following (but maybe I'm just blind).
Consider a graph $G$ with $k$ connected components $G_i$ of size $|G_i|$. The number of node p …
- 9,720
2
votes
3
answers
225
views
Random graphs defined by a set of tiles
Related to this question, which I asked at MSE, I'd like to ask this one here:
Consider a (large) graph $G$ and its multi-set of tiles $T$, i.e. the multi-set of its vertex-induced subgraphs, i.e. the …
4
votes
2
answers
2k
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Regular graph colorings
[Since I didn't get any feedback at MSE, I dare to post this question here, too.]
Call a coloring $C:V(G) \rightarrow \lbrace 1,\dots,n \rbrace$ of a graph $G$ regular when every vertex with color $c_ …
- 3,065
4
votes
3
answers
421
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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 n …
17
votes
3
answers
996
views
Primacy of arcs/arrows over vertices/objects
Freyd's Abelian Categories is the only textbook I know where the primacy of arrows over objects is taken seriously already in the axioms: there is no talk of objects at all. Only later one sees, that …
- 1,444
0
votes
0
answers
139
views
Graph theory: Closed neighourhoods and generalized clustering coefficients
The neighbourhood of node $v$ in graph $G$ is the subgraph of $G$ induced by all vertices adjacent to $v$.
The number of edges between neighbours divided by the number of pairs of neighbours is th …
- 9,720
3
votes
0
answers
83
views
Distance spectra of uniform tilings
Let a uniform tiling be defined by a vertex configuration $(n_1.n_2.\cdots.n_k)^m$, which is either spherical, Euclidean or hyperbolic. Assume that the tiling is vertex-transitive, especially that eac …
- 9,720
2
votes
1
answer
54
views
Uniform closure of a neighbourhood complex in the tritetragonal tiling
Consider a neighbourhood complex of eight vertices (red) with vertex configuration $(3.4)^3$ which gives rise to the tritetragonal tiling of the hyperbolic plane:
Not knowing if this complex can be u …
- 9,720