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...

Abstract. Topological indices are the numerical value associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. Graph theory is a delightful playground for the exploration of proof techniques in Discrete Mathematics and its results have applications in many areas of sciences. A graph is a ...

The idea of integral sum graphs was introduced by Harary (1994). A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f(u) + f(v) = f(w) for some node w in G. A tree is said to be a generalized star if it can be obtained from a star by extending each edge to...

Harary’s generalized Tic-Tac-Toe is an achievement game for polyominoes, where two players alternately put a stone on a grid board, and the player who first achieves a given polyomino wins the game. It is known whether the first player has a winning strategy in the generalized Tic-Tac-Toe for almost all polyominoes except the one called Snaky. GTTT(p, q) is an extension of the generalized Tic-T...