If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G;x) = α(G) ∑ k=0 skx k is the independence polynomial of G (Gutman and Harary, 1983). A graph G is very well-covered (Favaron, 1982) if it has no isolated vertices, its order equals 2α(G) and it is well-covered (i.e., all its maximal independent sets are of the same size, M...

The central problems in social sciences concern the social and psychological mechanisms and conditions required for the emergence and stability of human groups. The present article is dedicated to the problem of stability of human groups. We model human groups using local interacting systems of automaton with relations and reactions and using the structural balance theory. The ‘structural balan...

In a linear forest, each component is a path. The linear arboricity ~(G) of a graph G is defined in Harary [8] as the minimum number of linear forests whose union is G. This invariant first arose in a study [i0] of information retrieval in file systems. A quite similar covering invariant which is well known to the linear arboricity is the arboricity of a graph, which is defined as the minimum n...

If G and H are graphs, define the Ramsey number to be the least number p such that if the lines of the complete graph Kp are colored red and blue (say),either the red subgraph contains a copy of G or the blue subgraph contains H. Also set r(G) = r(G,G) ; these are called the diagonal Ramsey numbers. These definitions are taken from Chvátal and Harary [1J ; other terminology will follow Harary [...

Harary [8] calls a finite, simple graph G a sum graph if one can assign to each vi ∈ V (G) a label xi so that {vi, vj} ∈ E(G) iff xi + xj = xk for some k. We generalize this notion by replacing “xi + xj” with an arbitrary symmetric polynomial f(xi, xj). We show that for each f , not all graphs are “f -graphs”. Furthermore, we prove that for every f and every graph G we can transform G into an f...

Although natural languages are generally arbitrary in their mapping of forms to meanings, there are nevertheless some detectable biases in these mappings. For example, longer words tend to refer to meanings that are more conceptually complex (what we refer to as a complexity bias; Lewis, Sugarman, & Frank, 2014). The origins of this bias remain an open question, however. One hypothesis is that ...

A graph G is arbitrarily partitionable (AP for short) if for every partition (n1, n2, ..., np) of |V (G)| there exists a partition (V1, V2, ..., Vp) of V (G) such that each Vi induces a connected subgraph of G with order ni. If, additionally, k of these subgraphs (k ≤ p) each contains an arbitrary vertex of G prescribed beforehand, then G is arbitrarily partitionable under k prescriptions (AP+k...

In a given graph G = (V , E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring ofG if there exists a unique extension of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number. In this note, we study the chromatic n...

We show that several known Ramsey number inequalities can be extended to the setting of r-uniform hypergraphs. In particular, we extend Burr’s results on tree-star Ramsey numbers, providing exact evaluations for certain hypergraph Ramsey numbers. Then we turn our attention to proving a general multicolor hypergraph Ramsey number inequality from which generalizations of results due to Chvátal an...

We study Fräıssé limits for certain Fräıssé classes over infinite relational languages that behave similarly to those for finite languages. These include the Fräıssé classes of finite simplicial complexes, finite hypergraphs, and Sperner families on finite sets. With a natural choice of measure we show that almost all countable structures whose finite substructures lie in such a class are isomo...