  For a Urysohn space $X$ we define the regular diagonal degree
$\overline{\Delta}(X)$ of $X$ to be the minimal infinite cardinal $\kappa$ such
that $X$ has a regular $G_\kappa$-diagonal i.e. there is a family
$(U_\eta:\eta<\kappa)$ of open neighborhoods of $\Delta_X=\{(x,x)\in X^2:x\in
X\}$ in $X^2$ such that $\Delta_X = \bigcap_{\eta<\kappa} \overline{U}_\eta$.
  In this paper we show that if $X$ is a Urysohn space then: (1) $|X|\leq
2^{c(X)\cdot\overline{\Delta}(X)}$; (2) $|X|\leq 2^{\overline{\Delta}(X)\cdot
2^{wL(X)}}$; (3) $|X|\le wL(X)^{\overline{\Delta}(X)\cdot\chi(X)}$; and (4)
$|X|\le aL(X)^{\overline{\Delta}(X)}$; where $\chi(X)$, $c(X)$, $wL(X)$ and
$aL(X)$ are respectively the character, the cellularity, the weak Lindel\"of
number and the almost Lindel\"of number of $X$.
  The first inequality extends to the uncountable case Buzyakova's result that
the cardinality of a ccc-space with a regular $G_\delta$-diagonal does not
exceed $2^\omega$. It follows from (2) that every weakly Lindel\"of space with
a regular $G_\delta$-diagonal has cardinality at most $2^{2^\omega}$.
  Inequality (3) implies that when $X$ is a space with a regular
$G_\delta$-diagonal then $|X|\le wL(X)^{\chi(X)}$. This improves significantly
Bell, Ginsburg and Woods inequality $|X|\le 2^{\chi(X)wL(X)}$ for the class of
normal spaces with regular $G_\delta$-diagonals. In particular (3) shows that
the cardinality of every first countable space with a regular
$G_\delta$-diagonal does not exceed $wL(X)^\omega$.
  For the class of spaces with regular $G_\delta$-diagonals (4) improves Bella
and Cammaroto inequality $|X|\le 2^{\chi(X)\cdot aL(X)}$, which is valid for
all Urysohn spaces. Also, it follows from (4) that the cardinality of every
space with a regular $G_\delta$-diagonal does not exceed $aL(X)^\omega$.
