This answer uses also networkx,
else there is an overlap with the answer of fidbc .
First of all let us restate the question in a mathematical setting, fixing thus the key words for programming in sage.
Let $T$ be a tree with $n$ verices. Let $L=L(T)$ be the laplacian matrix of $T$. The laplace spectrum of $T$ is the spectrum of $L$,
$$\mu_1\ge \mu_2\ge \dots\ge \mu_{n-1}\ge\mu_n=0\ .$$
(Here, $0$ is always an eigenvalue with eigenvector $(1,1,\dots,1)$, since $T$ is connected. Its multiplicity is one.)
The second smallest value $\mu_{n-1}=a(T)$ is the
algebraic connectivity
of $T$. The biggest eigenvalue $\mu_1$ is the laplace spectral radius of $T$. Aka the norm of $L$.
The posted questions wants among all trees $T$ with $n=18$ vertices and diameter $d=5$ those $T$ with
$$\mu_1\mu_{n-1} = 1\ .$$
Then for each such tree $T$ also the Fiedler vector (see loc. cit.), i.e. the $L$--eigenvector corresponding to the algebraic connectivity $a(T)=\mu_{n-1}$.
There are $2015$ relevant trees with $n=18$ vertices and diameter $5$ .
sage: T18_5 = [ T for T in graphs.trees(18) if T.diameter() == 5 ]
sage: len( T18_5 )
2015
We select the needed subset:
T_LIST = []
# count = 0
for T in T18_5:
spec = T.laplacian_matrix().eigenvalues()
spec . sort()
# count += 1
# if count%100 == 0: print count
if spec[1] * spec[-1] == 1:
T_LIST.append( T )
Using networkx we can now ask for...
from networkx import algebraic_connectivity as a
from networkx import fiedler_vector as fiedler
for T in T_LIST:
Tx = T.networkx_graph()
print "a(T) = %s\nFiedler vector:\n%s\n" % ( a(Tx), fiedler(Tx) )
and get:
a(T) = 0.127016653793
Fiedler vector:
[-0.12469138 0.12469138 0.28566726 0.32723106 0.32723105 0.32723106
0.14283363 0.14283363 0.14283363 0.14283363 -0.28566726 -0.32723107
-0.32723105 -0.32723104 -0.14283363 -0.14283362 -0.14283364 -0.14283364]
a(T) = 0.14589803375
Fiedler vector:
[-0.15316861 0.15316861 0.22416617 0.26245832 0.22416619 0.26245836
0.22416616 0.26245831 0.22416616 0.26245831 -0.22416618 -0.26245834
-0.22416617 -0.26245833 -0.22416615 -0.26245829 -0.22416618 -0.26245834]
a(T) = 0.133108958618
Fiedler vector:
[-0.06494166 0.19176279 0.2688225 0.31009952 0.2688225 0.31009952
0.2688225 0.31009952 -0.25699018 -0.29645038 -0.29645038 -0.29645038
-0.29645038 -0.09103845 -0.1050172 -0.07491327 -0.07491327 -0.07491327]
This is all.
Note: Outside networkx we can also get valuable information,
for instance ask for the minimal polynomial of the algebraic_connectivity:
E = matrix.identity(18)
for T in T_LIST:
L = T.laplacian_matrix()
spec = L.eigenvalues()
spec . sort()
print " a(T) = %s with minpoly = %s" % ( spec[ 1], spec[ 1].minpoly() )
print "| L(T) | = %s with minpoly = %s" % ( spec[-1], spec[-1].minpoly() )
print "Fiedler vector:\n%s\n\n" % column_matrix(( L - spec[1]*E ).kernel().basis()[0])
# printed version of ( L - spec[1]*E ).kernel()
This gives:
a(T) = 0.1270166537925831? with minpoly = x^2 - 8*x + 1
| L(T) | = 7.872983346207417? with minpoly = x^2 - 8*x + 1
Fiedler vector:
[ 1]
[ -1]
[-2.290994448735806?]
[-2.624327782069139?]
[-2.624327782069139?]
[-2.624327782069139?]
[-1.145497224367903?]
[-1.145497224367903?]
[-1.145497224367903?]
[-1.145497224367903?]
[ 2.290994448735806?]
[ 2.624327782069139?]
[ 2.624327782069139?]
[ 2.624327782069139?]
[ 1.145497224367903?]
[ 1.145497224367903?]
[ 1.145497224367903?]
[ 1.145497224367903?]
a(T) = 0.1458980337503155? with minpoly = x^2 - 7*x + 1
| L(T) | = 6.854101966249684? with minpoly = x^2 - 7*x + 1
Fiedler vector:
[ 1]
[ -1]
[-1.463525491562421?]
[-1.713525491562421?]
[-1.463525491562421?]
[-1.713525491562421?]
[-1.463525491562421?]
[-1.713525491562421?]
[-1.463525491562421?]
[-1.713525491562421?]
[ 1.463525491562421?]
[ 1.713525491562421?]
[ 1.463525491562421?]
[ 1.713525491562421?]
[ 1.463525491562421?]
[ 1.713525491562421?]
[ 1.463525491562421?]
[ 1.713525491562421?]
a(T) = 0.1331089586175058? with minpoly = x^4 - 10*x^3 + 20*x^2 - 10*x + 1
| L(T) | = 7.512642352447085? with minpoly = x^4 - 10*x^3 + 20*x^2 - 10*x + 1
Fiedler vector:
[ 1]
[-2.952846325924191?]
[-4.139445001431823?]
[-4.775046463544424?]
[-4.139445001431823?]
[-4.775046463544424?]
[-4.139445001431823?]
[-4.775046463544424?]
[ 3.957245732488123?]
[ 4.564870950998888?]
[ 4.564870950998888?]
[ 4.564870950998888?]
[ 4.564870950998888?]
[ 1.401849112529162?]
[ 1.617099549550692?]
[ 1.153547507429801?]
[ 1.153547507429801?]
[ 1.153547507429801?]
In particular, we can identify the algebraic connectivity values as the algebraic numbers:
$$ 4-\sqrt{15}\ ,\qquad \frac 12(7-3\sqrt{5})\ ,\qquad \frac 12\left( 5+\sqrt7-\sqrt{28+10\sqrt7} \right)\ .$$
The laplacian spectral radius (also nom) is obtained by replacing the minus with a plus each time.
