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Results tagged with graph-theory
Search options questions only
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user 6094
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.
53
votes
1
answer
6k
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Why are there 1024 Hamiltonian cycles on an icosahedron?
Fix one edge $e$ of the graph (1-skeleton) of an icosahedron.
By a computer search, I found that there are 1024 Hamiltonian cycles that include $e$.
[But see edit below re directed vs. undirected!]
Wi …
- 151k
25
votes
2
answers
2k
views
Who first dubbed them "expander graphs"?
Expander graphs
("sparse graphs that have strong connectivity properties")
burst onto the mathematical scene around the millennium, but I have not
been successful in tracing the origin of
(a) the conc …
- 151k
24
votes
3
answers
2k
views
Gauss-Bonnet Theorem for Graphs?
One can define the Euler characteristic χ for a graph as the number of vertices minus the number of edges. Thus an $n$-cycle has $\chi = 0$ and $K_4$ has $\chi=-2$.
Is there an analog for the Gaus …
- 151k
24
votes
6
answers
3k
views
Shortest grid-graph paths with random diagonal shortcuts
Suppose you have a network of edges connecting
each integer lattice point
in the 2D square grid $[0,n]^2$
to each of its (at most) four neighbors, {N,S,E,W}.
Within each of the $n^2$ unit cells of thi …
- 151k
22
votes
2
answers
895
views
Is every 1-million-connected graph rigid in 3D?
It is an old result that every $6$-connected graph is rigid in $\mathbb{R}^2$:
Lovász, László, and Yechiam Yemini. "On generic rigidity in the plane." SIAM Journal on Algebraic Discrete Methods 3, no …
- 151k
20
votes
2
answers
1k
views
Erdős, Harary, Tutte's "dimension of graph": Progress in last 48 yrs?
I just ran across this delightful paper by an amazing triumvirate:
Paul Erdős, Frank Harary, and William Tutte. "On the dimension of a graph." Mathematika 12.118-122 (1965): 20.
(Cambridge link) …
- 151k
17
votes
3
answers
2k
views
Laplacians on graphs vs. Laplacians on Riemannian manifolds: $\lambda_2$?
A graph $G$ is connected if and only if
the second-largest eigenvalue $\lambda_2$ of
the Laplacian of $G$ is greater than zero.
(See, e.g.,
the Wikipedia article on algebraic connectivity.)
Is th …
- 151k
16
votes
3
answers
2k
views
Are infinite planar graphs still 4-colorable?
Imagine you have a finite number of "sites" $S$ in the positive quadrant
of the integer lattice $\mathbb{Z}^2$,
and from each site $s \in S$, one connects $s$ to every lattice point to which it
has a …
- 151k
16
votes
4
answers
1k
views
Squaring a square and discrete Ricci flow
Is this a theorem?
Every $3$-connected planar graph $G$ may be represented as
a tiling of a square by squares,
one square per node of $G$, with nodes connected in $G$
corresponding to tangent squares …
- 151k
15
votes
2
answers
752
views
Random noncrossing chords of a circle
Suppose you generate random chords of a circle, with endpoints selected uniformly over the circumference, rejecting any chord that crosses a previously generated chord.
The disk is then partitioned in …
- 151k
14
votes
1
answer
2k
views
Is every graph the center of some other graph?
The center of a graph $G$ is the set of vertices that minimize the largest
distance to vertices in $G$, e.g., in the graph below, that radius is $4$:
Define the center $C$ as the subgraph …
- 151k
14
votes
2
answers
730
views
Blinking graphs
For any simple graph $G$, assign its nodes a weight/bit of $0$ or $1$.
Call this a bit assignment for $G$.
Now, generate a new bit assignment as follows:
Each node $x$'s bit is replaced by $1$ if the …
- 151k
14
votes
2
answers
2k
views
A random walk on an infinite graph is recurrent iff ...?
Q. Is there a master theorem that can be used to determine whether or not
a simple random walk (choose a random neighboring vertex as the next step)
on a given infinite graph
leads to recurr …
- 151k
13
votes
2
answers
2k
views
Counting Hamiltonian cycles in $n \times n$ square grid
I wonder if anyone has counted these curves, either exactly or asymptotically?
Let $S_n$ be an $n \times n$ subset of $\mathbb{Z}^2$ consisting of $n^2$
lattice points: a lattice square.
Define a re …
- 151k
12
votes
1
answer
591
views
Characterizing graphs by their "walkers"
Let $G$ be a (large) graph and $W$ another (smaller) graph.
$W$ is what I call a walker.
Let me use "vertices" and "edges" for $G$ and
"nodes" and "arcs" for $W$.
$W$ has a distinguished node, its ce …
- 151k