The Geometry of some Fibonacci Identities in the Hosoya Triangle

In this paper we explore some Fibonacci identities that are interpreted geometrically in the Hosoya triangle. Specifically we explore a generalization of the Cassini and Catalan identities from a geometric point of view. We also extend some properties present in the Pascal triangle to the Hosoya triangle.Comment: There are 16 figures representing Fibonacci identities in the Hosoya triangl

Activity from matroids to rooted trees and beyond

The interior and exterior activities of bases of a matroid are well-known notions that for instance permit one to define the Tutte polynomial. Recently, we have discovered correspondences between the regions of gainic hyperplane arrangements and coloredlabeled rooted trees. Here we define a general activity theory that applies in particular to no-broken circuit (NBC) sets and labeled colored trees. The special case of activity \textsf{0} was our motivating case. As a consequence, in a gainic hyperplane arrangement the number of bounded regions is equal to the number of the corresponding colored labeled rooted trees of activity \textsf{0}.Comment: 7 Figure

Primes and composites in the determinant Hosoya triangle

In this paper, we look at numbers of the form $H_{r,k}:=F_{k-1}F_{r-k+2}+F_{k}F_{r-k}$. These numbers are the entries of a triangular array called the \emph{determinant Hosoya triangle} which we denote by ${\mathcal H}$. We discuss the divisibility properties of the above numbers and their primality. We give a small sieve of primes to illustrate the density of prime numbers in ${\mathcal H}$. Since the Fibonacci and Lucas numbers appear as entries in ${\mathcal H}$, our research is an extension of the classical questions concerning whether there are infinitely many Fibonacci or Lucas primes. We prove that ${\mathcal H}$ has arbitrarily large neighbourhoods of composite entries. Finally we present an abundance of data indicating a very high density of primes in ${\mathcal H}$.Comment: two figure

Maximizing the number of edges in optimal k-rankings

A k-ranking is a vertex k-coloring with positive integers such that if two vertices have the same color any path connecting them contains a vertex of larger color. The rank number of a graph is smallest k such that G has a k-ranking. For certain graphs G we consider the maximum number of edges that may be added to G without changing the rank number. Here we investigate the problem for G=P2k−1, C2k, Km1,m2,…,mt, and the union of two copies of Kn joined by a single edge. In addition to determining the maximum number of edges that may be added to G without changing the rank number we provide an explicit characterization of which edges change the rank number when added to G, and which edges do not