  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 $[-2, \infty )$.
