Avraham Goldstein

Balanced Abelian group-valued functions on directed graphs

We discuss functions from the edges and vertices of a directed graph to an Abelian group. A function is called balanced if the sum of its values along any cycle is zero. The set of all balanced functions forms an Abelian group under addition. We study this group in two cases: when we are allowed to walk against the direction of an edge taking the opposite value of the function and when we are not allowed to walk against the direction.

Groups of balanced labelings on graphs

We discuss functions from edges and vertices of an undirected graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced labelings. The set of balanced labelings forms an Abelian group. We study the structure of this group and the structure of two other groups, closely related to it: the subgroup of balanced labelings which consists of functions vanishing on vertices and the corresponding factor-group. This work is completely self-contained, except the algorithm for obtaining the 3-edge-connected components of an undirected graph, for which we make appropriate references to the literature.

On the structure of the group of balanced labelings on graphs

Let G = (V, E) be an undirected graph with possible multiple edges and loops (a multigraph). Let A be an Abelian group. In this work we study the following topics: 1) A function f:E → A is called balanced if the sum of its values along every closed truncated trail of G is zero. By a truncated trail we mean a trail without the last vertex. The set H(E, A) of all the balanced functions f: E → A is a subgroup of the free Abelian group A E of all functions from E to A. We give a full description of the structure of the group H(E, A), and provide an O(¦E¦)-time algorithm to construct a set of the generators of its cyclic direct summands. 2) A function g:V → A is called balanceable if there exists some f:E → A such that the sum of all the values of g and f along every closed truncated trail of G is zero. The set B(V, A) of all balanceable functions g:V → A is a subgroup of the free Abelian group A V of all the functions from V to A. We give a full description of the structure of the group B(V, A). 3) A function h:V ∪ E → A taking values on vertices and edges is called balanced if the sum of its values along every closed truncated trail of G is zero. The set W(V ∪ E, A) of all balanced functions h:V ∪ E → A is a subgroup of the free Abelian group A V∪E of all functions from V ∪ E to A. The group H(E, A) is naturally isomorphic to the subgroup of W(V ∪ E, A) consisting of all functions taking every vertex to 0. So we, abusing the notations, treat H(E, A) as that subgroup of W(V ∪ E, A).

Enumeration of balanced finite group valued functions on directed graphs

A group valued function on a graph is called balanced if the product of its values along any cycle is equal to the identity element of the group. We compute the number of balanced functions from the set of edges and vertices of a directed graph to a finite group considering two cases: when we are allowed to walk against the direction of an edge and when we are not allowed to walk against the edge direction. In the first case it appears that the number of balanced functions on edges and vertices depends on whether or not the graph is bipartite, while in the second case this number depends on the number of strong connected components of the graph. We study functions vanishing on each cycle of a directed graph.The number of such functions depends on whether or not the underlying undirected graph is bipartite.If every edge is one-way, then the number of such functions depends on the number of strongly connected components.

Stability of linear equations – algebraic approach

This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the coefficient matrix is obtained via experimentation. By giving the initial tolerance for the solution, the estimate of the admissible tolerance for the error matrix, and the error of the solution relative to the norm of the computed solution can be determined. The approach we use is standard properties of the Banach algebra on matrices equipped with the spectral norm.