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research
Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields
Authors
Swee Hong Chan
Henk D. L. Hollmann
Dmitrii V. Pasechnik
Publication date
1 January 2014
Publisher
'Elsevier BV'
Doi
Cite
View
on
arXiv
Abstract
A maximal minor
M
M
M
of the Laplacian of an
n
n
n
-vertex Eulerian digraph
Ξ
\Gamma
Ξ
gives rise to a finite group
Z
n
β
1
/
Z
n
β
1
M
\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M
Z
n
β
1
/
Z
n
β
1
M
known as the sandpile (or critical) group
S
(
Ξ
)
S(\Gamma)
S
(
Ξ
)
of
Ξ
\Gamma
Ξ
. We determine
S
(
Ξ
)
S(\Gamma)
S
(
Ξ
)
of the generalized de Bruijn graphs
Ξ
=
D
B
(
n
,
d
)
\Gamma=\mathrm{DB}(n,d)
Ξ
=
DB
(
n
,
d
)
with vertices
0
,
β¦
,
n
β
1
0,\dots,n-1
0
,
β¦
,
n
β
1
and arcs
(
i
,
d
i
+
k
)
(i,di+k)
(
i
,
d
i
+
k
)
for
0
β€
i
β€
n
β
1
0\leq i\leq n-1
0
β€
i
β€
n
β
1
and
0
β€
k
β€
d
β
1
0\leq k\leq d-1
0
β€
k
β€
d
β
1
, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime
p
p
p
and an
n
n
n
-cycle permutation matrix
X
β
G
L
n
(
p
)
X\in\mathrm{GL}_n(p)
X
β
GL
n
β
(
p
)
we show that
S
(
D
B
(
n
,
p
)
)
S(\mathrm{DB}(n,p))
S
(
DB
(
n
,
p
))
is isomorphic to the quotient by
β¨
X
β©
\langle X\rangle
β¨
X
β©
of the centralizer of
X
X
X
in
P
G
L
n
(
p
)
\mathrm{PGL}_n(p)
PGL
n
β
(
p
)
. This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS, and allows one to speculate upon a possibility to construct normal bases in the finite field
F
p
n
\mathbb{F}_{p^n}
F
p
n
β
from spanning trees in
D
B
(
n
,
p
)
\mathrm{DB}(n,p)
DB
(
n
,
p
)
.Comment: I+24 page
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Oxford University Research Archive
See this paper in CORE
Go to the repository landing page
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Last time updated on 18/05/2016