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Results tagged with spectral-graph-theory
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Questions related to the spectrum of graphs, defined using one of the possible variants of the discrete Laplace operator or Laplacian matrix. See https://en.wikipedia.org/wiki/Discrete_Laplace_operator
15
votes
Accepted
How can I prove that a particular family of graphs is integral?
$\def\CC{\mathbb{C}}$The specturm is integral.
The following trick is very useful in computing spectra of highly symmetric graphs. Let $G$ be a finite graph, let $\Gamma$ be a group of symmetries of …
- 156k
6
votes
Where does $2\sqrt{d-1}$ come from in Ramanujan graphs?
Heuristically, an expander $G$ looks locally like the $d$-regular tree $T$. Let $r$ be a positive real number and let $f_x$ be the function on the vertices of $T$ given by $f_x(y) = r^{d(x,y)}$. We ha …
- 156k
4
votes
Accepted
Number of trees with the same matching number
If $n=2m$, the answer is $2^{m-2} m! \binom{2m}{m} m^{m-2}$. Let $T_m = \sigma(n,m)$. We go through several transformations. We write $[n]$ for the set $\{ 1,2,\ldots, n \}$ and $[a,b]$ for $\{ a, a+1 …
- 156k