Hessenberg matrices and the Pell and Perrin numbers

AbstractIn this paper, we investigate the Pell sequence and the Perrin sequence and we derive some relationships between these sequences and permanents and determinants of one type of Hessenberg matrices

On the Mersenne sequence

From Binet’s formula of Mersenne sequence we get some properties for this sequence. Mersenne, Jacobsthal and Jacobsthal-Lucas sequences are considered in order to achieve some relations between them, sums and certain products involving terms of these sequences. We also present some results with matrices involving Mersenne numbers such as the generating matrix, tridiagonal matrices and circulant type matrices

On Certain Hessenberg Matrices Related with Linear Recurrences

In this paper, we present the permanents and determinants of some Hessenbergmatrices. Also, some special cases for permanents are given

Obituary - Richard Kenneth Guy, 1916-2020

PostprintPeer reviewe

Enumeration of Independent Sets in Graphs

An independent set is one of the most natural structures in a graph to focus on, from both a pure and applied perspective. In the realm of graph theory, and any concept it can represent, an independent set is the mathematical way of capturing a set of objects, none of which are related to each other. As graph theory grows, many questions about independent sets are being asked and answered, many of which are concerned with the enumeration of independent sets in graphs. We provide a detailed introduction to general graph theory for those who are not familiar with the subject, and then develop the basic language and notation of independent set theory before cataloging some of the history and major results of the field. We focus particularly on the enumeration of independent sets in various classes of graphs, with the heaviest focus on those defined by maximum and minimum degree restrictions. We provide a brief, specific history of this topic, and present some original results in this area. We then speak about some questions which remain open, and end the work with a conjecture for which we provide strong, original evidence. In the appendices, we cover all other necessary prerequisites for those without a mathematical background

Riordan graphs I : structural properties

In this paper, we use the theory of Riordan matrices to introduce the notion of a Riordan graph. The Riordan graphs are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other fami- lies of graphs. The Riordan graphs are proved to have a number of interesting (fractal) properties, which can be useful in creating computer networks with certain desirable features, or in obtaining useful information when designing algorithms to compute values of graph invariants. The main focus in this paper is the study of structural properties of families of Riordan graphs obtained from infinite Riordan graphs, which includes a fundamental decomposition theorem and certain conditions on Riordan graphs to have an Eulerian trail/cycle or a Hamiltonian cycle. We will study spectral properties of the Riordan graphs in a follow up paper