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Results tagged with graph-theory
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user 7410
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.
7
votes
Accepted
Matrix-tree theorem for inverse matrices
There is a formula as a sum of forests, i.e., collections of trees, for arbitrary minors of any size and row/column selection, for arbitrary matrices. So it can be applied for the numerator and the de …
2
votes
Accepted
Drunken X-mas polynomials for graphs
There is quite a bit of literature in statistical mechanics on spatially random permutations. The partition function essentially is $P_\Gamma(x)$ with $x=e^{-\alpha}$ for some temperature like paramet …
5
votes
Number of spanning forests in a graph
I am surprised no one mentioned the work of Alan Sokal and coworkers on precisely this issue
of weighted enumeration of spanning forests which is related to the $q\rightarrow 0$ limit of the
Potts mod …
- 2,175
1
vote
Accepted
How to prove that for the real stable characteristic polynomial $P=\Phi_T$ of a tree $T$, $P...
Let $B$ denote the $n\times n$ matrix $I_x-A$. Let $[n]=\{1,2,\ldots,n\}$.
For $I,J\subset [n]$ with the same number of elements, let $D_{I,J}$ denote the determinant of the matrix resulting from $B$ …
5
votes
Proofs where higher dimension or cardinality actually enabled much simpler proof?
An example in the spirit (in fact a generalization) of the answer by Sam T is the triviality of the $\phi^4$ quantum field theories from lattice approximations.
In dimension 5 or more this was done a …
3
votes
For what graph does the following algebraic property hold?
I don't know about the literature on your specific problem but I know this arises as a trivial subquestion of a more general one for which there is literature, most of it from the 19th century.
Let …
2
votes
Concepts of criticality in graph theory
I think the graph studied in my article "16,051 formulas for Ottaviani's invariant of cubic threefolds" with Christian Ikenmeyer and Gordon Royle fits the bill. In the paper we considered a hypergraph …
6
votes
The matrix tree theorem for weighted graphs
One can use the matrix-tree theorem with a suitable weight in order to compute the resultant of two polynomials in one variable. This is done in this 1908 article by Arthur Lee Dixon. There is also an …
2
votes
Enumeration of graphs arising in invariant theory
The number ${\rm dim}\ V_{n,v,e}$ is the number of linearly independent covariants of degree $v$ and weight $e$ of a binary form of degree $n$. This is the multiplicity of the irreducible module $Sym^ …