Related papers: Graphs associated with the map $x \mapsto x+x^{-1}…
In this paper we study the structure of the graphs associated with the iterations of the map $x \mapsto x+x^{-1}$ over finite fields of characteristic three.
In this paper we study the structure of the graphs associated with the iterations of the map $x \mapsto x+x^{-1}$ over finite fields of characteristic two. Formulas are given for the length of the cycles and the depth of the trees relying…
In this paper we describe the dynamics of certain rational maps of the form $k \cdot (x+x^{-1})$ over finite fields of odd characteristic.
The maps $x \mapsto ax^{2^k}+b$ defined over finite fields of characteristic two can be related to the duplication map over binary supersingular elliptic curves. Relying upon the structure of the group of rational points of such curves we…
In this paper we prove some results on the possible multiplicative orders of $\alpha + \alpha^{-1}$ when $\alpha$ is a non-zero element of a finite field of characteristic 2. The results of the paper rely on a previous investigation on the…
Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $n$ be a positive integer. In this paper, we study the digraph associated to the map $x\mapsto x^n h(x^{\frac{q-1}{m}})$, where $h(x)\in\mathbb{F}_q[x].$ We completely determine…
We investigate the structure of connected graphs, not necessarily locally finite, with infinitely many ends. On the one hand we study end-transitive such graphs and on the other hand we study such graphs with the property that the…
For polynomials and rational maps of fixed degree over a finite field, we bound both the average number of connected components of their functional graphs as well as the average number of periodic points of their associated dynamical…
We characterise all vertex-transitive finite connected graphs as essentially 5-connected or on a short list of explicit graph-classes. Our proof heavily uses Tutte-type canonical decompositions.
This note presents an approach to studying the iterates of a mapping whose restriction to the complement of a finite set is continuous and open. The main examples to which the approach can be applied are piecewise monotone mappings defined…
The problem of finding graph structure of functions commuting with a given function in terms of their functional graphs is considered. Structure of functional graphs of commuting functions is described. The problem is reduced to describing…
We define a graph structure associated in a natural way to finite fields that nevertheless distinguishes between different models of isomorphic fields.
The space of derivations of finite dimensional evolution algebras associated to graphs over a field with characteristic zero has been completely characterized in the literature. In this work we generalize that characterization by describing…
Let $\mathbb{F}_{q}$ be the finite field with $q$ elements. In this paper we will describe the dynamics of the map $f(X)=X(X^{q-1}-c)^{q+1},$ with $c\in\mathbb{F}_{q}^{\ast},$ over the finite field $\mathbb{F}_{q^2}$.
We study the relation between algebraic structures and Graph Theory. We have defined five different weighted digraphs associated to a finite dimensional algebra over a field in order to tackle important properties of the associated…
In this paper we study underlying graphs corresponding to a set of halving lines. We establish many properties of such graphs. In addition, we tighten the upper bound for the number of halving lines.
Each graph and choice of a commutative ring gives rise to an associated graphical group. In this article, we introduce and investigate graph polynomials that enumerate conjugacy classes of graphical groups over finite fields according to…
The directions of an infinite graph $G$ are a tangle-like description of its ends: they are choice functions that choose compatibly for all finite vertex sets $X\subseteq V(G)$ a component of $G-X$. Although every direction is induced by a…
We investigate graphs that have characteristic-dependent well-covered dimension and show how more of these graphs can be constructed from known ones.
This paper explains the periodicity of the Grover walk on finite graphs. We characterize the graphs to induce 2, 3, 4, 5-periodic Grover walk and obtain a necessary condition of the graphs to induce an odd-periodic Grover walk.