146 research outputs found
On polynomial digraphs
Let be a bivariate polynomial with complex coefficients. The
zeroes of are given a combinatorial structure by considering them
as arcs of a directed graph . This paper studies some relationship
between the polynomial and the structure of .Comment: 13 pages, 6 figures, See also http://www-ma2.upc.edu/~montes
Magic graphs and the faces of the Birkhoff polytope
Magic labelings of graphs are studied in great detail by Stanley and Stewart.
In this article, we construct and enumerate magic labelings of graphs using
Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes.
We define polytopes of magic labelings of graphs and digraphs. We give a
description of the faces of the Birkhoff polytope as polytopes of magic
labelings of digraphs.Comment: 9 page
Representation of Cyclotomic Fields and Their Subfields
Let \K be a finite extension of a characteristic zero field \F. We say
that the pair of matrices over \F represents \K if \K
\cong \F[A]/ where \F[A] denotes the smallest subalgebra of M_n(\F)
containing and is an ideal in \F[A] generated by . In
particular, is said to represent the field \K if there exists an
irreducible polynomial q(x)\in \F[x] which divides the minimal polynomial of
and \K \cong \F[A]/. In this paper, we identify the smallest
circulant-matrix representation for any subfield of a cyclotomic field.
Furthermore, if is any prime and \K is a subfield of the -th
cyclotomic field, then we obtain a zero-one circulant matrix of size
such that (A,\J) represents \K, where \J is the matrix with
all entries 1. In case, the integer has at most two distinct prime factors,
we find the smallest 0-1 companion-matrix that represents the -th cyclotomic
field. We also find bounds on the size of such companion matrices when has
more than two prime factors.Comment: 17 page
On the structure of the adjacency matrix of the line digraph of a regular digraph
We show that the adjacency matrix M of the line digraph of a d-regular
digraph D on n vertices can be written as M=AB, where the matrix A is the
Kronecker product of the all-ones matrix of dimension d with the identity
matrix of dimension n and the matrix B is the direct sum of the adjacency
matrices of the factors in a dicycle factorization of D.Comment: 5 page
Hamiltonian cycles in Cayley graphs of imprimitive complex reflection groups
Generalizing a result of Conway, Sloane, and Wilkes for real reflection
groups, we show the Cayley graph of an imprimitive complex reflection group
with respect to standard generating reflections has a Hamiltonian cycle. This
is consistent with the long-standing conjecture that for every finite group, G,
and every set of generators, S, of G the undirected Cayley graph of G with
respect to S has a Hamiltonian cycle.Comment: 15 pages, 4 figures; minor revisions according to referee comments,
to appear in Discrete Mathematic
On Hamilton decompositions of infinite circulant graphs
The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph).
Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}
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