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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.
27
votes
Accepted
Factorization of the characteristic polynomial of the adjacency matrix of a graph
Expanding on Richard's comment: let me rename your graph to $S$ and consider the adjacency matrix $A$ abstractly as a linear operator acting on the free vector space $\mathbb{C}[S]$ on (the vertices o …
- 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 …
- 4,713
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 …
- 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
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
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 $L$ …
CommunityBot
- 1
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
$$\text{tr } A^r = \sum_i \lambda_i^r$ …
- 118k
5
votes
Graph Theory: question regarding a class of digraph
If I understand you correctly, such a thing is generally called a partial function (from the vertex set to itself).
- 118k
3
votes
Spectral radius of a proper subgraph
There is a pretty straightforward counting argument which I give as Lemma 6 here. I don't think it is the standard argument, though.
- 118k
12
votes
How much linear algebra can be done with graphs?
Well, unlike the determinant, the eigenvalues of an integer matrix aren't integers, so I don't know how much to expect here as far as a direct combinatorial interpretation of any kind. However, there …
- 118k
7
votes
1
answer
528
views
Reference request: discrete harmonic functions and ends of graphs
Let $G$ be an infinite locally finite connected graph with finitely many ends. A real-valued function $f : G \to \mathbb{R}$ is harmonic if
$$f(v) = \frac{1}{d_v} \sum_{v \sim w} f(w)$$
where $v \ …
- 44.8k
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
13
votes
2
answers
1k
views
Combinatorial proof of (a special case of) the spectral theorem?
The spectral theorem for a real $n \times n$ symmetric matrix $A$ says that $A$ is diagonalizable with all eigenvalues real. If $A$ happens to have non-negative integer entries, it can be interpreted …
- 85.6k