5 research outputs found
Orbits of rotor-router operation and stationary distribution of random walks on directed graphs
The rotor-router model is a popular deterministic analogue of random walk. In
this paper we prove that all orbits of the rotor-router operation have the same
size on a strongly connected directed graph (digraph) and give a formula for
the size. By using this formula we address the following open question about
orbits of the rotor-router operation: Is there an infinite family of
non-Eulerian strongly connected digraphs such that the rotor-router operation
on each digraph has a single orbit?
It turns out that on a strongly connected digraph the stationary distribution
of the simple random walk coincides with the frequency of vertices in a rotor
walk. In this common aspect a rotor walk simulates a random walk. This gives
one similarity between two models on (finite) digraphs.Comment: 9 pages, 4 figures. To appear in Advances in Applied Mathematic
Multi-Eulerian tours of directed graphs
Not every graph has an Eulerian tour. But every finite, strongly connected
graph has a multi-Eulerian tour, which we define as a closed path that uses
each directed edge at least once, and uses edges e and f the same number of
times whenever tail(e)=tail(f). This definition leads to a simple
generalization of the BEST Theorem. We then show that the minimal length of a
multi-Eulerian tour is bounded in terms of the Pham index, a measure of
'Eulerianness'.Comment: 4 pages. Supersedes section 3 of arXiv:1502.04690v
CoEulerian graphs
We suggest a measure of "Eulerianness" of a finite directed graph and define
a class of "coEulerian" graphs. These are the graphs whose Laplacian lattice is
as large as possible. As an application, we address a question in chip-firing
posed by Bjorner, Lovasz, and Shor in 1991, who asked for "a characterization
of those digraphs and initial chip configurations that guarantee finite
termination." Bjorner and Lovasz gave an exponential time algorithm in 1992. We
show that this can be improved to linear time if the graph is coEulerian, and
that the problem is NP-complete for general directed multigraphs.Comment: 15 pages, to appear in Proc AMS. Main changes in v3: Removed the
section on multi-Eulerian tours, which will appear separately. Added Prop
2.13 on graphs that are both Eulerian and coEulerian. Added Table 3.1 on
computational complexit
Algorithmic aspects of rotor-routing and the notion of linear equivalence
We define the analogue of linear equivalence of graph divisors for the
rotor-router model, and use it to prove polynomial time computability of some
problems related to rotor-routing. Using the connection between linear
equivalence for chip-firing and for rotor-routing, we give a simple proof for
the fact that the number of rotor-router unicycle-orbits equals the order of
the Picard group. We also show that the rotor-router action of the Picard group
on the set of spanning in-arborescences can be interpreted in terms of the
linear equivalence
Random integral matrices: universality of surjectivity and the cokernel
For a random matrix of entries sampled independently from a fairly general
distribution in Z we study the probability that the cokernel is isomorphic to a
given finite abelian group, or when it is cyclic. This includes the probability
that the linear map between the integer lattices given by the matrix is
surjective. We show that these statistics are asymptotically universal (as the
size of the matrix goes to infinity), given by precise formulas involving zeta
values, and agree with distributions defined by Cohen and Lenstra, even when
the distribution of matrix entries is very distorted. Our method is robust and
works for Laplacians of random digraphs and sparse matrices with the
probability of an entry non-zero only n^{-1+epsilon}.Comment: 44 page