Counting degrees of vertices in near Goldbach graphs

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 $\frac{a+b}{2},\frac{\mathrm{\mid }a-b\mathrm{\mid }}{2}$ are either odd primes or $1$. A finite near Goldbach graph $G(n)$ has the vertex set $\{x\in 2\mathbb{N}:x\le 2n\}$ with the same adjacency rule. In this paper, we obtain two exact formulas for the degree of the even positive integer $x$ in $G(x\mathrm{/}2)$. We compute a function $\eta (x)=\prod _{p\mid x,p>2}\frac{p-1}{p-2}\frac{x{e}^{-0.183407}}{(\log x{)}^2}$ that approximates the degree of $x$ in $G(x\mathrm{/}2)$ 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.