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Results tagged with graph-theory
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user 11142
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.
6
votes
Accepted
Connected graphs that are not 2 connected
See
Kemkes, Graeme; Sato, Cristiane M.; Wormald, Nicholas, Asymptotic enumeration of sparse $2$-connected graphs, Random Struct. Algorithms 43, No. 3, 354–376 (2013). Zbl 1273.05127. PDF
and referen …
- 1,446
5
votes
Bounding number of k-cycles in a graph
Yes, this is addressed in the paper:
Rivin, Igor, Counting cycles and finite dimensional (L^{p}) norms, Adv. Appl. Math. 29, No. 4, 647-662 (2002). ZBL1013.05042.
- 96.4k
1
vote
The matrix tree theorem for weighted graphs
A somewhat different take on weighted trees is taken in this paper:
Jakobson, Dmitry; Rivin, Igor, Extremal metrics on graphs. I, Forum Math. 14, No. 1, 147-163 (2002). ZBL0995.05072.
In particular, …
- 96.4k
3
votes
A generously vertex transitive graph which is not Cayley?
It seems that any distance-transitive graph satisfies your condition (of course, distance-transitivity is much stronger), and you can find many examples (both finite and infinite) in Peter Cameron's n …
CommunityBot
- 1
1
vote
Embedding a graph in $\mathbb{R}^3$ with partial geometric information
There are a number of papers by Mike Treacy (I. Rivin is a co-author on some) which address this problem, but in a purely practical manner, using essentially the scheme proposed by Bullet51. Here is o …
- 96.4k
7
votes
Accepted
Length minimizing graphs between a finite set of points
This is the so-called Steiner Tree Problem.
- 96.4k
6
votes
Laplacian of an infinite graph and connected components
As Uri Bader points out, the infinite tree has an infinitely dimensional space of harmonic functions, so this is an answer to the philosophical part of the question: The way you prove that all harmoni …
- 96.4k
10
votes
Accepted
When does a row standardized adjacency matrix have a real spectrum?
If the adjacency matrix is $A,$ the "row-standardized" matrix is $DA$, where $D$ is a diagonal matrix all of whose diagonal entries are positive, so has a positive diagonal square root $D^{1/2}$. Now, …
4
votes
Accepted
Generalisation of Kuratowski's theorem
I am not sure which direction gives you trouble, but the second condition is necessary, by the Moore-Young theorem, or a somewhat weaker version thereof, as discussed by Greg Kuperberg here.
For the …
- 28k
4
votes
How many non-homeomorphic surfaces arise from these graphs?
It sounds like you are trying to enumerate ribbon graphs, in which case you might want to look at:
Do, Norman; Manescu, David, Quantum curves for the enumeration of ribbon graphs and hypermaps, Commu …
- 4,713
6
votes
Does there exist a notion of discrete riemannian metric on graph?
This circle of questions is studied in this old paper by D. Jakobson and I. Rivin.
- 96.4k
3
votes
Expected spectral radius for a sparse Erdős-Rényi binary matrix with a certain density
If you mean that $A$ is the adjacency matrix of an Erdos-Renyi random graph, then the question has been studied, and your conjecture is false (but just barely). See
Krivelevich, Michael; Sudakov, Ben …
- 96.4k
14
votes
A random walk on an infinite graph is recurrent iff ...?
This is a huge subject, but the best introductory reference remains:
Doyle, Peter G.; Snell, J.Laurie, Random walks and electric networks, The Carus Mathematical Monographs, 22. Washington, D. C.: Th …
- 96.4k
1
vote
Accepted
References studying properties of a graph which are stable under finite perturbation
The references for these can be found in Doyle and Snell's deathless classic (which is available for free on arXiv.org. ). Section 2.4 is particularly a propos.
Doyle, Peter G.; Snell, J.Laurie, Rand …
- 96.4k
2
votes
Number of non-equivalent graph embeddings
Peter Heinig's answer is excellent, but here are some further remarks:
Under the "ambient isotopy" definition, there are infinitely many classes of embeddings (because the mapping class group is inf …
- 96.4k