We introduce a new tool, called the orbit automaton, that describes the action of an automaton group G on the subtrees corresponding to the orbits of G on levels of the tree. The connection between G and the groups generated by the orbit automata is used to find elements of infinite order in certain automaton groups for which other methods failed to work.

The Strahler number or Horton-Strahler number of a tree, originally introduced in geophysics, has a surprisingly rich theory. We sketch some milestones in its history, and its connection to arithmetic expressions, graph traversing, decision problems for context-free languages, Parikh’s theorem, and Newton’s procedure for approximating zeros of differentiable functions.

Kontsevich conjectured that the number of zeros over the field Fq of a certain polynomial QG associated with the spanning trees of a graph G is a polynomial function of q. We show the connection between this conjecture, the Matrix-Tree Theorem, and orthogonal geometry. We verify the conjecture in certain cases, such as the complete graph, and discuss some modifications and extensions.

Given a graph G with vertex set V(G) and edge E(G), for the bijective function f(V(G))?{1,2,?,|V(G)|}, associated weight of an xy?E(G) under f is w(xy)=f(x)+f(y). If all edges have pairwise distinct weights, called edge-antimagic labeling. A path P in vertex-labeled said to be rainbow x?y if every two xy,x?y??E(P) it satisfies w(xy)?w(x?y?). The antimagic labeling there exists vertices x,y?V(G)...

In this paper, the concept of fraction tree is introduced. This has profound consequences both in mathematical sub-disciplines and other branches science technology. we shall witness connection with that Fibonacci sequence relate terms continued fractions.

Fractional share models are used to reason about how multiple actors share ownership of resources. We examine the decidability and complexity of reasoning over the “tree share” model of Dockins et al. using first-order logic, or fragments thereof. We pinpoint a connection between the basic operations on trees union t, intersection u, and complement and countable atomless Boolean algebras, allow...

Finn V. Jensen Frank Jensen Department of Mathematics and Computer Science Aalborg University Fredrik Bajers Vej 7E, DK-9220 Aalborg st, Denmark E-mail: [email protected], [email protected] Abstract The paper deals with optimality issues in connection with updating beliefs in networks. We address two processes: triangulation and construction of junction trees. In the rst part, we give a simple algor...

We study Sushchansky p-groups introduced in [Sus79]. We recall the original definition and translate it into the language of automata groups. The original actions of Sushchansky groups on p-ary tree are not level-transitive and we describe their orbit trees. This allows us to simplify the definition and prove that these groups admit faithful level-transitive actions on the same tree. Certain br...

The distributed algorithm for a multicast connection set-up, based on the &cheapest insertion' heuristic, is reviewed. The multicast routing problem is translated into a Steiner tree problem in point-to-point networks where nodes have only a limited knowledge about the network. A solution is proposed in which the time complexity and the amount of information exchanged between network nodes are ...

The standard Bayesian Information Criterion (BIC) is derived under regularity conditions which are not always satisfied in the case of graphical models with hidden variables. In this paper we derive the BIC for the binary graphical tree models where all the inner nodes of a tree represent binary hidden variables. This provides an extension of a similar formula given by Rusakov and Geiger for na...