Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertic...

An edge uv in a graph Γ is directionally 2-signed (or, (2, d)signed) by an ordered pair (a, b), a, b ∈ {+, −}, if the label l(uv) = (a, b) from u to v, and l(vu) = (b, a) from v to u. Directionally 2-signed graphs are equivalent to bidirected graphs, where each end of an edge has a sign. A bidirected graph implies a signed graph, where each edge has a sign. We extend a theorem of Sriraj and Sam...

ABSTRACT In a signed graph, each link is labeled with either a positive or a negative sign. This is particularly appropriate to model polarized systems. Such a graph can be characterized through the notion of structural balance, which relies on the partitioning of the graph into internally solidary but mutually hostile subgroups. In this work, we show that signed graphs can be used to model and...

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...

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 ...