In this paper we prove that no signed graph on the complete graph Kp, p ≥ 6, is graceful, and we also give a characterization of graceful signed graphs on Kp, p ≤ 5. This implies that there is no subset A of cardinality p ≥ 6 from the set {0, 1, . . . , (p 2 ) − n}, n ≤ 1 2 ( p 2 ) , such that each element of the set {1, 2, . . . , n} occurs exactly twice and each element of the remaining set {...

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...

Motivation Want to learn a combinatorial parameter of a graph: the maximum matching size the independence number α(G), the minimum vertex cover size, the minimum dominating set size Krzysztof Onak – Sublinear Graph Approximation Algorithms – p. 2/32 Motivation Want to learn a combinatorial parameter of a graph: the maximum matching size the independence number α(G), the minimum vertex cover siz...

Let $G=(V,E)$ be a simple graph. A set $Dsubseteq V$ is adominating set of $G$ if every vertex in $Vsetminus D$ has atleast one neighbor in $D$. The distance $d_G(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$G$. An $(u,v)$-path of length $d_G(u,v)$ is called an$(u,v)$-geodesic. A set $Xsubseteq V$ is convex in $G$ ifvertices from all $(a, b)$-geodesics belon...

An important property of chordal graphs is that these graphs are characterized by existence of perfect elimination orderings on their vertex sets. In this paper, we generalize the notion of perfect elimination orderings to signed graphs, and give a characterization for graphs admitting such orderings, together with characterizations restricted to some subclasses and further properties of those ...

A signed graph is a simple graph where each edge receives a sign positive or negative. Such graphs are mainly used in social sciences where individuals represent vertices friendly relation between them as a positive edge and enmity as a negative edge. In signed graphs, we define these relationships (edges) as of friendship ("+" edge) or hostility ("-" edge). A 2-path product signed graph [Formu...