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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.
0
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0
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188
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Graph theory meta-question
If property (P) holds for perfect graphs and almost all graphs does it hold for all graphs?
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3
votes
2
answers
412
views
Graphs with a unique transmission value
If $G$ is a graph with distance function $d(x,y)$ between vertices, the transmission of a vertex $x \in v(G)$ is defined as $\sigma_{x}=\sum_{y \neq x}{d(x,y)}$. I want to know if there is a known cha …
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0
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2
answers
866
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Reference for "almost all graphs have diameter 2"
The property in the title is well-known. I am trying to find an original reference to its first appearance in print. The 4th edition of Graphs & Digraphs by Chartrand and Lesniak lists this as Theorem …
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6
votes
1
answer
342
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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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3
votes
1
answer
617
views
Toroidality testing
Is there a standard test for the recognition of toroidal graphs? I have been using the Boyer-Myrvold planarity algorithm, which has a MATLAB and C++ implementation, and I would like to know if there …
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1
vote
1
answer
179
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Have you come across this kind of "degree" concept?
Let $A \subseteq V(G)$ be a set of vertices in a graph $G$ and let $v \in V(G)$ be some vertex. Define $d_{A}(v)$ as the number of neighbours of $v$ inside $A$.
Now suppose you have a graph whose ver …
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1
vote
2
answers
223
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Is number of quasi-kernels NP-hard?
A quasi-kernel in a directed graph D is an independent subset of vertices $S$ so that for every $v \in V(D)-S$ either $v->s$ for some $s \in S$ or $v->w->s$ for some $w \in V(D)-S, s \in S$.
Equivale …
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4
votes
1
answer
281
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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 …
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0
votes
2
answers
1k
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Confused about orbits
I am trying to apply the main theorem of this paper to a certain kind of graph and keep getting confused. The theorem uses $rank(Aut\Gamma)$ which is defined as "the number of $Aut \Gamma$ orbits on t …
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1
vote
1
answer
390
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almost-bipartite nearly-isolated subgraphs
I am looking for examples/families of graphs with the following (maybe vague-sounding at first) property: the graph $G$ has a relatively large subgraph $B$ such that $B$ is bipartite-plus-a-few-edges …
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6
votes
1
answer
438
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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:
$$
\sum_{i=1}^{n}{d_{i}^{2}} \leq e(\frac{2e}{ …
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7
votes
1
answer
506
views
Full-rank factorization of the graph Laplacian
Is there a combinatorially meaningful full-rank factorization of the Laplacian matrix of a graph?
The usual factorization $L=BB^{T}$, where $B$ is an oriented incidence matrix, is full-rank if and on …
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8
votes
1
answer
312
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A conjecture about strongly regular graphs
Let $G \neq K_{v}$ be a $(v,k,\lambda,\mu)$ strongly regular graph. After perusing through Brouwer's tables of parameters I have formed the conjecture $$\lambda-\mu \leq \frac{k}{2}.$$
So far I have …
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0
votes
1
answer
324
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Graphs with circulant distance matrices
The cycle has this property. For instance, the distance matrix for a 6-cycle is:
$A=\begin{bmatrix}
0 & 1 & 2 & 3 & 2 & 1 \\\\
1 & 0 & 1 & 2 & 3 & 2 \\\\
…
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5
votes
1
answer
296
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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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