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Results tagged with graph-theory
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user 4556
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
1
answer
261
views
Existence of a nice subset of edges in $k-$regular simple graphs?
Let $G=(V,E)$ be a finite simple $k-$regular graph ($k\geq 1$). Does $G$ necessarily
contain a subset $E'\subset E$ of edges such that only isolated edges and cycles occur as connected components in $ …
- 18.4k
7
votes
2
answers
810
views
Efficient computation of a vertex-partition for graphs
A finite simple connected graph $\Gamma$ with vertices $V(\Gamma)$ has a
partition of its vertices into (at most) two subsets defined as follows:
Given a spanning tree $T\subset \Gamma$, chose a fun …
- 18.4k
1
vote
Maximum number of shortest-paths
I do not see the interest of introducing $r$. The number of shortest paths from $s$ to $t$
passing through $r$ is the product of the number of shortest paths from $s$ to $r$ times
the number of short …
- 18.4k
2
votes
Ways to "regularize" a graph
Given a finite simple graph $\Gamma$ with vertices $V$, one can consider the Cayley graph
of the group generated by transpositions of vertices corresponding to edges.
This group is the symmetric group …
- 18.4k
4
votes
"Antipodal" maps on regular graphs?
Gerhard's example is sharp, the answer is "yes" for $D=2$. Indeed, consider the complementary
graph and apply the answer to question
Existence of a nice subset of edges in $k-$regular simple graphs? …
- 18.4k
1
vote
How many graphs with given average degree and average number of outgoing nodes?
I do not understand what your outgoing edges are but the number of undirected graphs with
$N$ labeled vertices and average degree $d$ (which you call $z_1$ but $d$ seems more natural
to me)
is approx …
- 18.4k
4
votes
0
answers
127
views
Counting vertex-permutations of a finite tree which rip all edges
Given a finite tree $T$ with $n$ vertices labelled $1,\dots,n$, we say that a permutation $\sigma$ of $1,\dots,n$ rips all edges if $\{\sigma(i),\sigma(j)\}$
is never an edge for every edge $\{i,j\}$ …
- 18.4k
2
votes
Similarity of weighted graphs
If all weights of edges are strictly positive, interpreting them as lengths and considering the Gromov-Hausdorff distance (see http://en.wikipedia.org/wiki/Gromov%E2%80%93Hausdorff_convergence) is per …
- 18.4k
5
votes
Spectral graph theory: Interpretability of eigenvalues and -vectors
(Intended as a comment to Chris Godsil's answer, but too long.)
One way of making the last statement in Chris Godsil's answer precise is as follows: Consider a planar graph which is $3-$edge connecte …
- 18.4k
1
vote
Counting connected fundamental domains of actions on Cayley graphs
An inefficient way for enumerating all connected fundamental domains is by remarking that they are
all contained (up to translation) in the ball of radius at most $(l+1)/2$ (with respect to word lengt …
- 18.4k
8
votes
2
answers
1k
views
Maximal number of directed edges in suitable simple graphs on $n$ vertices without directed ...
We consider the class $C$ of directed simple (no multiple edges) graphs having the property that every vertex is reachable by a directed path from every other vertex.
Given an integer $k$, what is th …
- 18.4k
2
votes
Weighted Regular Graphs
Denoting by $A$ the $n\times n$ adjaceny matrix (with loops contributing $1$
on the diagonal) of a finite graph $\Gamma$ with $n$ vertices, a necessary condition for $A$ to be weighted-regular is the …
- 18.4k
3
votes
Integral positive definite quadratic forms and graphs
I have studied a closely related problem (but never published the results and my notes
are messy).
Consider a set $\mathcal S$ of roots in a simply laced root system. Associate to $\mathcal S$
the gra …
- 18.4k
2
votes
Cyclic Permutations - but not what you think
An elementary way to describe the moduli spaces alluded in the reply of Tilman is as follows:
Glue the sides of two oriented polygons $P,P'$ with $n$ sides, counterclockwise labeled from 1 to $n$ by a …
- 18.4k
1
vote
Algorithm to solve Sokoban-like game on graphs - move chips from one set of vertices to another
Not an answer but a reformulation of Sokoban in terms of oriented graphs: A Sokoban-problem of order $k$ on a finite directed
graph with vertices $V$ and directed edges $E$ (with $(s,t)$ denoting an e …
- 18.4k