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Results tagged with graph-theory
Search options questions only
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user 1737
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.
33
votes
10
answers
6k
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Is the empty graph a tree?
This is a boring, technical question that I stumbled upon while making a contribution to Sage. I would still like to hear a constructive answer so hopefully the question does not get closed.
The que …
- 3,463
19
votes
3
answers
2k
views
A generalization of the triangle counting problem for simple weighted graphs
One nice identity is that $$\operatorname{tr}(A^3)/6$$ counts the number of triangles of a graph with adjacency matrix $A$. It also implies that triangle counting in a graph can be performed in sub-cu …
- 3,463
19
votes
4
answers
1k
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Minimal graphs with a prescribed number of spanning trees
As it's long ago since Erdős died and MathOverflow is the second best alternative to him (for discussing personal problems), I'd like to start a fruitful discussion about the following problem that I …
- 3,463
14
votes
1
answer
887
views
Tutte polynomials, graph complements and degree sequences
Harary and Akiyama asked whether there exists a non self-complementary (SC) graph $G$ having the same chromatic polynomial as its complement.
It was later shown that there indeed exist such graphs a …
- 3,463
11
votes
1
answer
968
views
Is every matching of the hypercube graph extensible to a Hamiltonian cycle
Given that $Q_d$ is the hypercube graph of dimension $d$ then it is a known fact (not so trivial to prove though) that given a perfect matching $M$ of $Q_d$ ($d\geq 2$) it is possible to find another …
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8
votes
4
answers
1k
views
Prime numbers $p$ not of the form $ab + bc + ac$ $(0 < a < b < c )$ (and related questions)
If we ask which natural numbers n are not expressible as $n = ab + bc + ca$ ($0 < a < b < c$) then this is a well known open problem. Numbers not expressible in such form are called Euler's "numeru …
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8
votes
1
answer
328
views
Maximal class of simple graphs of order $n$ with mutually distinct numbers of spanning trees
This problem in some ways related to this post.
Let $A_n$ be the set of all integers $x$ such that there exist a connected simple graph of order $n$ having precisely $x$ spanning trees. Study the …
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8
votes
2
answers
349
views
Isomorphism problem on the class of induced subgraphs of a hypercube
A problem that I am currently studying translates to the problem of deciding whether two induced subgraphs of the hypercube $Q_k$ are isomorphic.
Now it feels to me that this class of graphs is "too …
- 3,463
7
votes
1
answer
403
views
Induced subgraphs of small strongly regular graphs
Consider a strongly regular graph $G$ with parameters $(76,30,8,14).$ Hoffman's bound tells us that $\overline{G}$ has an independent set of size at most $4$ and its not hard to see there are indeed $ …
- 3,463
7
votes
3
answers
1k
views
Randomly contracting edges of a graph - expected number of vertices?
Let $G'$ be a graph obtained from $G$ after contracting each edge with probability $p$. Let $n = |V(G)|, e = |E(G)|$.
I would like to compute (or at least obtain a lower bound) for $E[|V(G')|]$ in te …
- 3,463
7
votes
1
answer
821
views
(The) missing Moore graph(s) - uniqueness
In the related literature one often sees the phrase "The missing Moore graph" which (to me) tacitly implies that the missing Moore graph (if exists) is unique.
Is there a result of this type or is or …
- 3,463
6
votes
1
answer
685
views
Probability that a random edge coloring of the complete graph is proper
This is a repost of this math.se question that I am posting here since it received no attention there.
What is the probability that a random edge coloring of $K_n$ with
$m \geq n$ colors resul …
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6
votes
2
answers
854
views
Minimal graphs of prescribed girth and chromatic number
The well known result of Erdős, states that
Given integers $g > 2$ and $k > 1$ there exist a graph $G$ with $\chi(G) \geq k$ and girth at least $g.$
What I am wondering is
When can we expect …
- 3,463
5
votes
0
answers
319
views
Graphs with many positive eigenvalues of their distance matrix
Let $G$ be a simple connected graph $D(G)$ its distance matrix and $n_{+}(G), n_{-}(G)$ the number of positive and negative eigenvalues of $D(G)$ respectively.
We call a graph $G$ optimistic if $n_{+ …
- 3,463
5
votes
2
answers
780
views
Neat results from algebraic graph theory
Studying algebraic graph theory, I've stumbled across a wide range of results that I found pretty stunning and also useful. I'd like to share them here and ask for your favorite result from this area? …