Signed graphs provide models for investigating balance in connection with various kinds of social relations. Since empirical social networks always involve uncertainty because of errors due to measurement, imperfect observation or sampling, it is desirable to incorporate uncertainty into signed graph models. We introduce a stochastic signed graph and investigate the properties of some indices o...

A certain signed adjacency matrix of the hypercube, which Hao Huang used last year to resolve sensitivity conjecture, is closely related unique, 4-cycle free, 2-fold cover hypercube. We develop a framework in this connection natural first example relationship between group labeled matrices with few eigenvalues, and combinatorially interesting covering graphs. In particular, we define two-eigenv...

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 {...

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...