Let
$R$
be a ring and
$Z(R)$
be the set of all zero-divisors of
$R$
. The total graph of
$R$
, denoted by
$T(\Gamma (R))$
is a graph with all elements of
$R$
as vertices, and two distinct vertices
$x, y\in R$
are adjacent if and only if
$x+ y\in Z(R)$
. Let the regular graph of
$R$
,
$\mathrm{Reg} (\Gamma (R))$
, be the induced subgraph of
$T(\Gamma (R))$
on the regular elements of
$R$
. In 2008, Anderson and Badawi proved that the girth of the total graph and the regular graph of a commutative ring are contained in the set
$\{ 3, 4, \infty \} $
. In this paper, we extend this result to an arbitrary ring (not necessarily commutative). We also prove that if
$R$
is a reduced left Noetherian ring and
$2\not\in Z(R)$
, then the chromatic number and the clique number of
$\mathrm{Reg} (\Gamma (R))$
are the same and they are
${2}^{r} $
, where
$r$
is the number of minimal prime ideals of
$R$
. Among other results, we show that if
$R$
is a semiprime left Noetherian ring and
$\mathrm{Reg} (R)$
is finite, then
$R$
is finite.