Regular character-graphs whose eigenvalues are greater than or equal to -2

Let $G$ be a finite group and $\mathrm{Irr}(G)$ be the set of all complex irreducible characters of $G$. The character-graph $\Delta(G)$ associated to $G$, is a graph whose vertex set is the set of primes which divide the degrees of some characters in $\mathrm{Irr}(G)$ and two distinct primes $p$ and $q$ are adjacent in $\Delta(G)$ if the product $pq$ divides $\chi(1)$, for some $\chi\in\mathrm{Irr}(G)$. Tong-viet posed the conjecture that if $\Delta(G)$ is $k$-regular for some integer $k\geqslant 2$, then $\Delta(G)$ is either a complete graph or a cocktail party graph. In this paper, we show that his conjecture is true for all regular character-graphs whose eigenvalues are in the interval $[-2, \infty )$.