Counting degrees of vertices in near Goldbach graphs

Abstract

A near Goldbach graph is a simple undirected graph whose vertex set consists of all positive even integers and there is an edge between two vertices $a,b$ if and only if $\frac{a+b}{2}, \frac{|a-b|}{2}$ are either odd primes or $1$. A finite near Goldbach graph $G(n)$ has the vertex set $\{x\in 2\mathbb{N}\, :\, x\leq 2n\}$ with the same adjacency rule. In this paper, we obtain two exact formulas for the degree of the even positive integer $x$ in $G(x/2)$. We compute a function $η(x)=\prod\limits_{p\mid x,\, p>2} \frac{p-1}{p-2}\, \frac{xe^{-0.183407}}{(\log\, x)^2}$ that approximates the degree of $x$ in $G(x/2)$ for a large even positive integer $x$. Finally, we introduce the concept of a nearly independent set of events and show that if the set of divisibility events for a large even integer $x$ is nearly independent, then $x$ can be expressed as the sum of two odd primes.