Multiplicative graphs ant its applications to the equation $n - \varphi(n)=c$

We study the following general problem. Given natural number $c$ and a pair of multiplicative functions $f,g$. The question is to find the number of solutions of the equation
$$ f(n)\,-\,g(n)\,=\,c. $$
Under some conditions to the solutions of this equation and to the functions $f,g$ (in particular, $f(n) > g(n)$ for $n > 1$), we prove that the number of such solutions does not exceed $c^{\,1 - \varepsilon}$. Next, for the number $J(c)$ of solutions of the equation
$$ n - \varphi(n) = c, $$
we find the followingformula:
$$ J(c)\,=\,G(c + 1) + O(c^{\,3/4 + o(1)}). $$
Here $G(k)$ denotes the number of representations of $k$ by the sum of two primes. In order to obtain such results, we use so-called multiplicative graphs.