67,621 research outputs found
Directed strongly walk-regular graphs
We generalize the concept of strong walk-regularity to directed graphs. We
call a digraph strongly -walk-regular with if the number of
walks of length from a vertex to another vertex depends only on whether
the two vertices are the same, adjacent, or not adjacent. This generalizes also
the well-studied strongly regular digraphs and a problem posed by Hoffman. Our
main tools are eigenvalue methods. The case that the adjacency matrix is
diagonalizable with only real eigenvalues resembles the undirected case. We
show that a digraph with only real eigenvalues whose adjacency matrix
is not diagonalizable has at most two values of for which can
be strongly -walk-regular, and we also construct examples of such
strongly walk-regular digraphs. We also consider digraphs with nonreal
eigenvalues. We give such examples and characterize those digraphs for
which there are infinitely many for which is strongly
-walk-regular
Algebraic and combinatorial aspects of sandpile monoids on directed graphs
The sandpile group of a graph is a well-studied object that combines ideas
from algebraic graph theory, group theory, dynamical systems, and statistical
physics. A graph's sandpile group is part of a larger algebraic structure on
the graph, known as its sandpile monoid. Most of the work on sandpiles so far
has focused on the sandpile group rather than the sandpile monoid of a graph,
and has also assumed the underlying graph to be undirected. A notable exception
is the recent work of Babai and Toumpakari, which builds up the theory of
sandpile monoids on directed graphs from scratch and provides many connections
between the combinatorics of a graph and the algebraic aspects of its sandpile
monoid.
In this paper we primarily consider sandpile monoids on directed graphs, and
we extend the existing theory in four main ways. First, we give a combinatorial
classification of the maximal subgroups of a sandpile monoid on a directed
graph in terms of the sandpile groups of certain easily-identifiable subgraphs.
Second, we point out certain sandpile results for undirected graphs that are
really results for sandpile monoids on directed graphs that contain exactly two
idempotents. Third, we give a new algebraic constraint that sandpile monoids
must satisfy and exhibit two infinite families of monoids that cannot be
realized as sandpile monoids on any graph. Finally, we give an explicit
combinatorial description of the sandpile group identity for every graph in a
family of directed graphs which generalizes the family of (undirected)
distance-regular graphs. This family includes many other graphs of interest,
including iterated wheels, regular trees, and regular tournaments.Comment: v2: Cleaner presentation, new results in final section. Accepted for
publication in J. Combin. Theory Ser. A. 21 pages, 5 figure
A family of mixed graphs with large order and diameter 2
A mixed regular graph is a connected simple graph in which each vertex has both a fixed outdegree (the same indegree) and a fixed undirected degree. A mixed regular graphs is said to be optimal if there is not a mixed regular graph with the same parameters and bigger order.
We present a construction that provides mixed graphs of undirected degree qq, directed degree View the MathML sourceq-12 and order 2q22q2, for qq being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to View the MathML source9q2-4q+34 the defect of these mixed graphs is View the MathML source(q-22)2-14.
In particular we obtain a known mixed Moore graph of order 1818, undirected degree 33 and directed degree 11 called Bosák’s graph and a new mixed graph of order 5050, undirected degree 55 and directed degree 22, which is proved to be optimal.Peer ReviewedPostprint (author's final draft
- …