Let $G$ be a finite group and $\mathrm{Irr}(G)$ be the set of all complex irreducible characters of $G$. The character-graph $Δ(G)$ 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 $Δ(G)$ if the product $pq$ divides $χ(1)$, for some $χ\in\mathrm{Irr}(G)$. Tong-viet posed the conjecture that if $Δ(G)$ is $k$-regular for some integer $k\geqslant 2$, then $Δ(G)$ 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 $[-2, \infty )$.