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Results tagged with graph-theory
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user 22051
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.
21
votes
1
answer
4k
views
What have simplicial complexes ever done for graph theory?
(I am asking in a somewhat tongue-in-cheek fashion, of course, but nevertheless...)
Are there examples of results in "classical" [*] graph theory that have
been achieved by using simplicial comp …
- 6,962
11
votes
5
answers
2k
views
Are all almost regular graphs obvious?
Let the maximum and minimum degress of a graph be denoted (as usual) by $\Delta$ and $\delta$ respectively.
A graph is almost regular if $\Delta-\delta=1$.
Now, here is a simple way to generate …
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9
votes
1
answer
414
views
Coherence between different ranking methods of a graph's vertices
Given a (connected) graph $G$ it is natural to want to rank its vertices, with the more "central" vertices ranked higher.
Two natural ways of doing it are:
By the degrees.
By the entries in a Perro …
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8
votes
1
answer
335
views
Spectral lower bounds on the diameter of a graph
There is such a bound, due to Mohar and McKay, using the second-smallest eigenvalue of the Laplacian $\lambda_{2}$:
$$Diam \geq \lceil \frac{4}{n\lambda_{2}} \rceil.$$
This bound is very elegant but …
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8
votes
1
answer
312
views
A conjecture about strongly regular graphs
Let $G \neq K_{v}$ be a $(v,k,\lambda,\mu)$ strongly regular graph. After perusing through Brouwer's tables of parameters I have formed the conjecture $$\lambda-\mu \leq \frac{k}{2}.$$
So far I have …
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7
votes
1
answer
506
views
Full-rank factorization of the graph Laplacian
Is there a combinatorially meaningful full-rank factorization of the Laplacian matrix of a graph?
The usual factorization $L=BB^{T}$, where $B$ is an oriented incidence matrix, is full-rank if and on …
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7
votes
7
answers
3k
views
Efficient Hamiltonian cycle algorithms for graph classes
Generally speaking, finding a Hamiltonian cycle is NP-Hard and so tough. But if $G=L(H)$ is the line graph of $H$, then we can reduce the problem of finding a Hamiltonian cycle in $G$ to finding an Eu …
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6
votes
1
answer
342
views
Hamiltonian cycles in power-graphs
I've stumbled across a short note from 1993 where a nice question was asked: Suppose you take a graph with vertices $\{1,2,\ldots,n\}$ and connect $i,j$ by an edge if and only if $i+j$ is a $k$th powe …
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6
votes
1
answer
236
views
Hamiltonicity criteria for sparse graphs
Given a sparse graph, how can one go about proving that it is Hamiltonian? (Assuming it actually is, of course).
There are three main classes of criteria for Hamiltonicity that I am aware of:
Dirac …
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6
votes
1
answer
438
views
What is/are the best bound/s on the sum of squares of degrees in a graph?
Let $G$ be a graph with degrees $d_{1},\ldots,d_{n}$. I am interested in upper bounds on
$$
\sum_{i=1}^{n}{d_{i}^{2}}.
$$
An example is de Caen's bound:
$$
\sum_{i=1}^{n}{d_{i}^{2}} \leq e(\frac{2e}{ …
- 6,962
5
votes
1
answer
282
views
Duration and critical groups order in sandpile models and chip firing games
The famous chip firing game (which is closely related to sandpile models) goes like this:
Place chips at the vertices of a graph. REPEATEDLY: If a vertex $v$ of
degree $d_{v}$ has at least $d_{ …
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5
votes
1
answer
296
views
Graphs with constant edge imbalance
The imbalance of an edge $(u,v) \in E(G)$ of a graph $G$ is defined as $|d(u)-d(v)|$ ($d$ being, as usual the degree). (This concept was introduced by Albertson in 1997)
I'm interested in the set of …
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5
votes
1
answer
1k
views
How many distinct eigenvalues does a random graph have?
It is well-known that a random graph a.e. has diameter 2. It is also well-known that the number of distinct eigenvalues of a graph is at least the diameter plus one.
But what is known about the expec …
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4
votes
1
answer
281
views
If a graph invariant is NP-Hard, is its "deck ratio" NP-Hard as well?
This question is inspired by the Graph Reconstruction Conjecture. Suppose that $\psi$ is some graph invariant and that it is NP-Hard. There is a plethora of examples, of course. Now define $D_{\psi}(G …
- 6,962
4
votes
0
answers
242
views
Sets of spreads in graphs
Let $G$ be a graph. A $k$-spread is a set of cliques of order $k$ which partition the vertex set (so $k|n$, where $n$ is the number of vertices).
A partial $k$-resolution of $G$ is a set of pairwise …
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