Given a vertex u ∈ V of a graph Γ = (V,E), the (local) proper polynomials constitute a sequence of orthogonal polynomials, constructed from the so-called u-local spectrum of Γ. These polynomials can be thought of as a generalization, for all graphs, of the distance polynomials for the distance-regular graphs. The (local) adjacency polynomials, which are basically sums of proper polynomials, wer...

Graph orientation is a well-studied area of graph theory. A proper orientation of a graph G = (V,E) is an orientationD of E(G) such that for every two adjacent vertices v and u, d D (v) 6= d D (u) where d D (v) is the number of edges with head v in D. The proper orientation number of G is defined as −→χ (G) = min D∈Γ max v∈V (G) d D (v) where Γ is the set of proper orientations of G. We have χ(...

Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let $(T_s)_{s\in \mathbb{R}}$ be globally definable one parameter family of $C^2$-hypersurfaces $\mathbb{R}^n$. Upon defining notion generalized critical value for such we show that functions $s \to |K(s)|$ and $s\to K(s)$, respectively total absolute Gauss-Kronecker curvature $T_s$, are continuous in any neighb...

The two-player game of Nim on graphs is played on a regular graph with positively weighted edges by moving alternately from a fixed starting vertex to an adjacent vertex, decreasing the weight of the incident edge to a strictly smaller non-negative integer. The game ends when a player is unable to move since all edges incident with the vertex from which the player is to move have weight zero. I...

In the perpetual gossiping problem, introduced by Liestman and Richards, information may be generated at any time and at any vertex of a graph G; adjacent vertices can communicate by telephone calls. We define Wk(G) to be the minimum w such that, placing at most k calls each time unit, we can ensure that every piece of information is known to every vertex within w time units of its generation. ...

It is well-known that a polynomial f(z) = adz(1 + o(1)) can be conjugated by a holomorphic map φ to w 7→ w in a neighbourhood of in nity. This map φ is called a Böttcher coordinate for f near in nity. In this paper we construct a Böttcher type coordinate for compositions of a ne mappings and polynomials, a class of mappings rst studied in [9]. As an application, we prove that if h is a ne and c...

Although cellular automata have been widely studied as a class of the spatio temporal systems, very few investigators have studied how to identify the CA rules given observations of the patterns. A solution using a polynomial realization to describe the CA rule is reviewed in the present study based on the application of an orthogonal least squares algorithm. Three new neighbourhood detection m...

The paper is devoted to studying the 1D viscous Burgers equation controlled by an external force. It is assumed that the initial state is essentially bounded, with no decay condition at infinity, and the control is a trigonometric polynomial of low degree with respect to the space variable. We construct explicitly a control space of dimension 11 that enables one to steer the system to any neigh...

Given an undirected multigraph G = (V;E) and a requirement function r : 0 V 2 1 ! Z + (where 0 V 2 1 is the set of all pairs of vertices and Z + is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the local edge-connectivity and vertex-connectivity b e t ween every pair x; y 2 V become at least r (x; y) and two, respectively. ...