Nonnegative integral subset representations of integer sets

dominating subset and representation graph on topological spaces

Let a topological space. An intersection graph on a topological space , which denoted by ‎ , is an undirected graph which whose vertices are open subsets of and two vertices are adjacent if the intersection of them are nonempty. In this paper, the relation between topological properties of  and graph properties of ‎  are investigated. Also some classifications and representations for the graph ...

On Optimal Subset Representations of Integer Sets

1 7 Fe b 20 03 Generalized additive bases , König ’ s lemma , and the Erdős - Turán conjecture ∗

Let A be a set of nonnegative integers. For every nonnegative integer n and positive integer h, let rA(n, h) denote the number of representations of n in the form n = a1 + a2 + · · · + ah, where a1, a2, . . . , ah ∈ A and a1 ≤ a2 ≤ · · · ≤ ah. The infinite set A is called a basis of order h if rA(n, h) ≥ 1 for every nonnegative integer n. Erdős and Turán conjectured that lim sup n→∞ rA(n, 2) = ...

Finite groups admitting a connected cubic integral bi-Cayley graph

A graph   is called integral if all eigenvalues of its adjacency matrix  are integers.  Given a subset $S$ of a finite group $G$, the bi-Cayley graph $BCay(G,S)$ is a graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid sin S, xin G}$.  In this paper, we classify all finite groups admitting a connected cubic integral bi-Cayley graph.

Additive Bases with Many Representations

In additive number theory, the set A of nonnegative integers is an asymptotic basis of order 2 if every sufficiently large integer can be written as the sum of two elements of A . Let rA (n) denote the number of representations of n in the form n = a+a', where a, a' eA and a < a' . An asymptotic basis A of order 2 is minimal if no proper subset of A is an asymptotic basis of order 2 . Erdös and...