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Results tagged with graph-theory
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user 290
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.
5
votes
Graph Theory: question regarding a class of digraph
If I understand you correctly, such a thing is generally called a partial function (from the vertex set to itself).
- 118k
9
votes
Accepted
How many non-isomorphic graphs of 50 vertices and 150 edges
The simplest guess one could make is $\frac{1}{50!} { {50 \choose 2} \choose 150}$. That is, we first count the number of labeled such graphs, then assume that most of them have trivial automorphism g …
- 118k
7
votes
Graph with group structure?
I can at least propose a definition. To my mind, the categorically best behaved category of graphs is the category of presheaves on $\{ \bullet \rightrightarrows \bullet \}$ ("directed multigraphs"); …
- 118k
3
votes
"Homomorphism fingerprint" for graphs
For graphs $G, H$, let $G \times H$ denote the graph with vertex set $V(G) \times V(H)$ and such that $(g_0, h_0)$ and $(g_1, h_1)$ are connected by an edge iff $g_0$ and $g_1$ are connected by an edg …
- 118k
5
votes
When Do a Few Eigenvectors of Graph Laplacians Not Determine the Graph?
Everything I'm about to say depends on knowing the eigenvectors and eigenvalues exactly (in terms of algebraic numbers).
Lemma: Let $L$ be an integer square matrix and $\lambda$ an eigenvalue of $L$ …
- 118k
10
votes
Accepted
Is there an "adjacency matrix" for weighted directed graphs that captures the weights?
It looks like in your definition the weight of a path is the sum of the weights of its edges. In many combinatorial applications a natural definition of the weight of a path is the product of the wei …
- 118k
2
votes
Which integer recurrence relations can be formulated as counting walks on a graph?
Okay, so it's not quite a duplicate because I guess you're asking about initial conditions as well. The generating functions of the sequences $a_n$ which have this property are called $\mathbb{N}$-re …
- 118k
29
votes
Accepted
Number of closed walks on an $n$-cube
Yes (assuming a closed walk can repeat vertices). For any finite graph $G$ with adjacency matrix $A$, the total number of closed walks of length $r$ is given by
$$\text{tr } A^r = \sum_i \lambda_i^r$ …
- 118k
4
votes
1
answer
232
views
Reference request: "unoriented composition" in generalized categories
I'm looking for a generalized notion of category (really of symmetric multicategory) which, roughly speaking, doesn't make a distinction between sources and targets. Each "morphism" in such a category …
- 118k
22
votes
3
answers
1k
views
Why are Dynkin diagrams characterized by their eigenvalues?
The Dynkin diagrams An, Dn, E6,
E7, E8 can be characterized among finite simple connected
graphs by the property that their eigenvalues (that is, the eigenvalues of their adjacency matrices) all have …
- 118k
9
votes
What (fun) results in graph theory should undergraduates learn?
The (finite, simple) graphs with the property that their adjacency matrices have spectral radius less than $2$ are precisely the simply laced Dynkin diagrams $A_n, D_n, E_6, E_7, E_8$. Similarly, the …
3
votes
Spectral radius of a proper subgraph
There is a pretty straightforward counting argument which I give as Lemma 6 here. I don't think it is the standard argument, though.
- 118k
7
votes
1
answer
528
views
Reference request: discrete harmonic functions and ends of graphs
Let $G$ be an infinite locally finite connected graph with finitely many ends. A real-valued function $f : G \to \mathbb{R}$ is harmonic if
$$f(v) = \frac{1}{d_v} \sum_{v \sim w} f(w)$$
where $v \ …
- 118k
2
votes
Reconstructing an ordering of a multiset from its consecutive submultisets
No "palindromic sandwich" is reconstructible. By this I mean an ordering of the form $aba'$ where $a'$ is $a$ reversed and $a$ is at least the length of the peek of the initial segment you get. This …
- 118k
2
votes
Spectral properties of Cayley graphs
I know at least one special case where your second question makes sense. If $G$ is a compact group, it has a category $\text{Rep}(G)$ of finite-dimensional unitary representations which break up into …
- 118k