82 research outputs found
A contribution to the connections between Fibonacci Numbers and Matrix Theory
We present a lovely connection between the Fibonacci numbers and the sums of
inverses of triangular matrices, namely, a number is the sum of
the entries of the inverse of an triangular
matrix iff is an integer between and . Corollaries
include Fibonacci identities and a Fibonacci type result on determinants of
family of (1,2)-matrices.Comment: 7 pages, 2 figure
Arrangements Of Minors In The Positive Grassmannian And a Triangulation of The Hypersimplex
The structure of zero and nonzero minors in the Grassmannian leads to rich
combinatorics of matroids. In this paper, we investigate an even richer
structure of possible equalities and inequalities between the minors in the
positive Grassmannian. It was previously shown that arrangements of equal
minors of largest value are in bijection with the simplices in a certain
triangulation of the hypersimplex that was studied by Stanley, Sturmfels, Lam
and Postnikov. Here we investigate the entire set of arrangements and its
relations with this triangulation. First, we show that second largest minors
correspond to the facets of the simplices. We then introduce the notion of
cubical distance on the dual graph of the triangulation, and study its
relations with the arrangement of t-th largest minors. Finally, we show that
arrangements of largest minors induce a structure of partially ordered sets on
the entire collection of minors. We use the Lam and Postnikov circuit
triangulation of the hypersimplex to describe a 2-dimensional grid structure of
this poset
Conjectured bounds for the sum of squares of positive eigenvalues of a graph
A well known upper bound for the spectral radius of a graph, due to Hong, is
that . It is conjectured that for connected graphs , where denotes the sum of the squares of the
positive eigenvalues. The conjecture is proved for various classes of graphs,
including bipartite, regular, complete -partite, hyper-energetic, and
barbell graphs. Various searches have found no counter-examples. The paper
concludes with a brief discussion of the apparent difficulties of proving the
conjecture in general
Equal Entries in Totally Positive Matrices
We show that the maximal number of equal entries in a totally positive (resp.
totally nonsingular) matrix is (resp.
)). Relationships with point-line incidences in the plane,
Bruhat order of permutations, and completability are also presented. We
also examine the number and positionings of equal minors in a
matrix, and give a relationship between the location of
equal minors and outerplanar graphs.Comment: 15 page
Arrangements of equal minors in the positive
Abstract. We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with sorted sets, which earlier appeared in the context of alcoved polytopes and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the Eulerian number. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the weakly separated sets. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the positive Grassmannian and the associated cluster algebra. Résumé. Il s’agit des arrangements des mineurs égaux dans les matrices totalement positives. Plus précisément, nous aimerions étudier la structure des égalités et inégalités possibles entre les mineurs. Nous montrons que les arrangements des mineurs égaux de plus grande valeur sont en bijection avec les ensembles triés, qui auparavant apparaissaient dans le cadre de polytopes alcôve et bases de Gröbner. Arrangements maximales de ce format correspondent aux simplexes de la triangulation alcôve de la hypersimplex, et le nombre de ces arrangements est égal au nombre eulérien. D’autre part, nous conjecturons et prouvons dans des cas nombreux que les arrangements des mineurs égaux de plus petite valeur sont notamment les ensembles faiblement séparés. Ces ensembles faiblement séparés, initialement introduites par Leclerc et Zelevinsky, sont liés à la Grassmannienne positive et l’algèbre cluster
Eigenvalue Pairing in the Response Matrix for a Class of Network Models with Circular Symmetry
We consider the response matrices in certain weighted networks that display a circular symmetry. It had been observed empirically that these exhibit several paired (multiplicity two) eigenvalues. Here, this pairing is explained analytically for a version of the model more general than the original. The exact number of necessarily paired eigenvalues is given in terms of the structure of the model, and the special structure of the eigenvectors is also described. Examples are provided
Arrangements of equal minors in the positive Grassmannian
We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with sorted sets, which earlier appeared in the context of alcoved polytopes and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the Eulerian number. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the weakly separated sets. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated cluster algebra
Arrangements Of Minors In The Positive Grassmannian And a Triangulation of The Hypersimplex
The structure of zero and nonzero minors in the Grassmannian leads to rich combinatorics of matroids. In this paper, we investigate an even richer structure of possible equalities and inequalities between the minors in the positive Grassmannian. It was previously shown that arrangements of equal minors of largest value are in bijection with the simplices in a certain triangulation of the hypersimplex that was studied by Stanley, Sturmfels, Lam and Postnikov. Here we investigate the entire set of arrangements and its relations with this triangulation. First, we show that second largest minors correspond to the facets of the simplices. We then introduce the notion of cubical distance on the dual graph of the triangulation, and study its relations with the arrangement of t-th largest minors. Finally, we show that arrangements of largest minors induce a structure of partially ordered sets on the entire collection of minors. We use the Lam and Postnikov circuit triangulation of the hypersimplex to describe a 2-dimensional grid structure of this poset
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