We present a decomposition formula for the weighted zeta function of an irregular covering of a graph by its weighted L-functions. Moreover, we give a factorization of the weighted zeta function of an (irregular or regular) covering of a graph by equivalence classes of prime, reduced cycles of the base graph. As an application, we discuss the structure of balanced coverings of signed graphs.

A signed graph is called net regular if the sum of the signs of every edge incident to each vertex is constant. Graphs that admit a signing making them net regular are called net regularizable. In this paper, net regular signed trees are studied, including general properties, conditions for a tree to be net regularizable, and generating functions.

Algorithms for search of communities in networks usually consist discrete variations of links. Here we discuss a flow method, driven by a set of differential equations. Two examples are demonstrated in detail. First is a partition of a signed graph into two parts, where the proposed equations are interpreted in terms of removal of a cognitive dissonance by agents placed in the network nodes. Th...

An extension of balance notion from the theory of signed graphs to the case of finite sets systems is presented. For a finite set T , a subset S ⊆ T and a family F of subsets of T we denote by δm (S|F) respectively δM (S|F) the minimum/maximum number of changes (addition or deletion of elements), without repetition, which transforms S into a set from F . We are especially interested in the part...

We prove that for every dimension s and every number n of points, there exists a point-set Pn,s whose -weighted unanchored L∞ discrepancy is bounded from above by C(b)/n1/2−b independently of s provided that the sequence = {γk} has ∑∞ k=1 γ a k < ∞ for some (even arbitrarily large) a. Here b is a positive number that could be chosen arbitrarily close to zero and C(b) depends on b but not on s o...