We show that the Smith normal form of a skew‐symmetric D‐optimal design of order
n
≡
2
(
mod
4
) is determined by its order. Furthermore, we show that the Smith normal form of such a design can be ...written explicitly in terms of the order
n, thereby proving a recent conjecture of Armario. We apply our result to show that certain
D‐optimal designs of order
n
≡
2
(
mod
4
) are not equivalent to any skew‐symmetric
D‐optimal design. We also provide a correction to a result in the literature on the Smith normal form of
D‐optimal designs.
k-Primitivity of digraphs Beasley, LeRoy B.; Mousley, Sarah
Linear algebra and its applications,
05/2014, Volume:
449
Journal Article
Peer reviewed
Open access
Let D be a directed graph (digraph) on n vertices. The digraph D is said to be primitive if for some m, between any ordered pair of vertices of D there is a directed walk of length m from the first ...vertex to the other. Here our focus is a generalization of primitivity, called k-primitivity, where k-arc-colorings of digraphs are considered. Let kmax(n) be the maximum k for which there exists a k-coloring of some strong n-tournament that is k-primitive. We show that (n−12)⩽kmax(n)<(n2)−⌈n4⌉.
In this article, we define the binary codes of tournament matrices in the class T(R) and give unique construction algorithms for matrices which have minimum and maximum binary codes. By introducing a ...generating algorithm with an order we show that all matrices in class T(R) can be sorted uniquely between matrices with minimum and maximum binary codes.
Let B
denote the Brualdi-Li matrix of order 2m, and let ρ
= ρ(B
) denote the spectral radius
of the Brualdi-Li Matrix. Then
where m > 2, e = 2.71828 · · · ,
and
In this paper we derive new properties complementary to an 2
n
×2
n
Brualdi-Li tournament matrix
B
2
n
. We show that
B
2
n
has exactly one positive real eigenvalue and one negative real eigenvalue ...and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of
B
2
n
is also determined. Related results obtained in previous articles are proven to be corollaries.
Boolean rank of upset tournament matrices Brown, David E.; Roy, Scott; Lundgren, J. Richard ...
Linear algebra and its applications,
05/2012, Volume:
436, Issue:
9
Journal Article
Peer reviewed
Open access
The Boolean rank of an m×n(0,1)-matrix M is the minimum k for which matrices A and B exist with M=AB, A is m×k, B is k×n, and Boolean arithmetic is used. The intersection number of a directed graph D ...is the minimum cardinality of a finite set S for which each vertex v of D can be represented by an ordered pair (Sv,Tv) of subsets of S such that there is an arc from vertex u to vertex v in D if and only if Su∩Tv≠Ø. The intersection number of a digraph is equal to the Boolean rank of its adjacency matrix. Using this fact, we show that the intersection number of an upset tournament, equivalently, the Boolean rank of its adjacency matrix, is equal to the number of maximal subpaths of certain types in its upset path.
Extending partial tournaments Beasley, LeRoy B.; Brown, David E.; Reid, K. Brooks
Mathematical and computer modelling,
07/2009, Volume:
50, Issue:
1
Journal Article
Peer reviewed
Open access
Let
A
be a
(
0
,
1
,
∗
)
-matrix with main diagonal all 0’s and such that if
a
i
,
j
=
1
or
∗
then
a
j
,
i
=
∗
or 0. Under what conditions on the row sums, and or column sums, of
A
is it possible to ...change the
∗
’s to 0’s or 1’s and obtain a tournament matrix (the adjacency matrix of a tournament) with a specified score sequence? We answer this question in the case of regular and nearly regular tournaments. The result we give is best possible in the sense that no relaxation of any condition will always yield a matrix that can be so extended.
Let
A
be a square (0, 1)-matrix. Then
A
is a Hall matrix provided it has a nonzero permanent. The Hall exponent of
A
is the smallest positive integer
k
, if such exists, such that
A
k
is a Hall ...matrix. The Hall exponent has received considerable attention, and we both review and expand on some of its properties. Viewing
A
as the adjacency matrix of a digraph, we prove several properties of the Hall exponents of line digraphs with some emphasis on line digraphs of tournament (matrices).
Among all the ways one might define
f(
A) for a square complex matrix
A and a given function
f:
C→
C
, the notion of a
primary matrix function is perhaps the most useful and natural. Using only basic ...conceptual properties of primary matrix functions, we consider whether
f(
A) can have rank one, and whether, for a given
B, there is a unique
A such that
f(
A)=
B.
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