#20228: Spectral radius of graphs
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Reporter: vdelecroix | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-7.1
Component: graph theory | Resolution:
Keywords: | Merged in:
Authors: Vincent Delecroix | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/vdelecroix/20228 | dd2dd58d732460eeb713112306a02d37a6750db9
Dependencies: | Stopgaps:
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Description changed by vdelecroix:
Old description:
> There are no current way to compute the spectral radius of a graph (the
> Perron-Frobenius eigenvalue of its adjacency matrix). We provide a method
> `spectral_radius` that provides an interval of floating approximations.
>
> It is much faster than previously available methods (where the exact
> methods are unusable because of characteristic polynomial computation)
> {{{
> sage: G = digraphs.RandomDirectedGNM(10,40)
> sage: %timeit max(G.adjacency_matrix().charpoly().roots(AA,
> multiplicities=False))
> 100 loops, best of 3: 6.23 ms per loop
> sage: %timeit G.spectral_radius()
> 1000 loops, best of 3: 178 µs per loop
> }}}
> And for a larger graph
> {{{
> sage: G = digraphs.RandomDirectedGNM(400, 6000)
> sage: %timeit max(G.adjacency_matrix().charpoly().roots(AA,
> multiplicities=False))
> 1 loop, best of 3: 4.63 s per loop
> sage: %timeit G.spectral_radius()
> 10 loops, best of 3: 13 ms per loop
> }}}
> The new method `spectral_radius` can be used for much larger graph as
> soon as the spectral graph is large enough to ensure the convergence.
New description:
There are no current way to compute the spectral radius of a graph (the
Perron-Frobenius eigenvalue of its adjacency matrix). We provide a method
`spectral_radius` that provides an interval of floating approximations.
It is much faster than previously available methods (where the exact
methods are unusable because of characteristic polynomial computation)
{{{
sage: G = digraphs.RandomDirectedGNM(10,40)
sage: %timeit max(G.adjacency_matrix().charpoly().roots(AA,
multiplicities=False))
100 loops, best of 3: 6.23 ms per loop
sage: %timeit G.spectral_radius()
1000 loops, best of 3: 178 µs per loop
}}}
And for a larger graph
{{{
sage: G = digraphs.RandomDirectedGNM(400, 6000)
sage: %timeit max(G.adjacency_matrix().charpoly().roots(AA,
multiplicities=False))
1 loop, best of 3: 4.63 s per loop
sage: %timeit G.spectral_radius()
10 loops, best of 3: 13 ms per loop
}}}
The new method `spectral_radius` can be used for much larger graph as soon
as the spectral gap is large enough to ensure the convergence.
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Ticket URL: <http://trac.sagemath.org/ticket/20228#comment:2>
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