1,741 research outputs found
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
On the Cayley digraphs that are patterns of unitary matrices
A digraph D is the pattern of a matrix M when D has an arc ij if and only if
the ij-th entry of M is nonzero. Study the relationship between unitary
matrices and their patterns is motivated by works in quantum chaology and
quantum computation. In this note, we prove that if a Cayley digraph is a line
digraph then it is the pattern of a unitary matrix. We prove that for any
finite group with two generators there exists a set of generators such that the
Cayley digraph with respect to such a set is a line digraph and hence the
pattern of a unitary matrix
Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number
The quantum chromatic number of a graph is sandwiched between its
chromatic number and its clique number, which are well known NP-hard
quantities. We restrict our attention to the rank-1 quantum chromatic number
, which upper bounds the quantum chromatic number, but is
defined under stronger constraints. We study its relation with the chromatic
number and the minimum dimension of orthogonal representations
. It is known that . We
answer three open questions about these relations: we give a necessary and
sufficient condition to have , we exhibit a class of
graphs such that , and we give a necessary and
sufficient condition to have . Our main tools are
Kochen-Specker sets, collections of vectors with a traditionally important role
in the study of noncontextuality of physical theories, and more recently in the
quantification of quantum zero-error capacities. Finally, as a corollary of our
results and a result by Avis, Hasegawa, Kikuchi, and Sasaki on the quantum
chromatic number, we give a family of Kochen-Specker sets of growing dimension.Comment: 12 page
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