17 research outputs found
Sandpile groups and spanning trees of directed line graphs
We generalize a theorem of Knuth relating the oriented spanning trees of a
directed graph G and its directed line graph LG. The sandpile group is an
abelian group associated to a directed graph, whose order is the number of
oriented spanning trees rooted at a fixed vertex. In the case when G is regular
of degree k, we show that the sandpile group of G is isomorphic to the quotient
of the sandpile group of LG by its k-torsion subgroup. As a corollary we
compute the sandpile groups of two families of graphs widely studied in
computer science, the de Bruijn graphs and Kautz graphs.Comment: v2 has an expanded section on deletion/contraction for directed
graphs, and a more detailed proof of Theorem 2.3. To appear in Journal of
Combinatorial Theory A
Automorphisms of necklaces and sandpile groups
We introduce a group naturally acting on aperiodic necklaces of length
with two colours using the 1--1 correspondences between aperiodic necklaces and
irreducible polynomials over the field \F_2 of two elements. We notice that
this group is isomorphic to the quotient group of non-degenerate circulant
matrices of size over that field modulo a natural cyclic subgroup. Our
groups turn out to be isomorphic to the sandpile groups for a special sequence
of directed graphs.Comment: 12 pages, several tables, no picture