  Let $G$ be a finite simple connected graph on $[n]$ and $R = K[x_1, \ldots,
x_n]$ the polynomial ring in $n$ variables over a field $K$. The edge ideal of
$G$ is the ideal $I(G)$ of $R$ which is generated by those monomials $x_ix_j$
for which $\{i, j\}$ is an edge of $G$. In the present paper, the possible
tuples $(n, {\rm depth} (R/I(G)), {\rm reg} (R/I(G)), \dim R/I(G), {\rm deg} \
h(R/I(G)))$, where ${\rm deg} \ h(R/I(G))$ is the degree of the $h$-polynomial
of $R/I(G)$, arising from Cameron--Walker graphs on $[n]$ will be completely
determined.
