  Let $G$ be the circulant graph $C_n(S)$ with $S \subseteq \{1, 2, \dots,
\lfloor \frac{n}{2} \rfloor\}$, and let $I(G)$ denote the edge ideal in the
polynomial ring $R=\mathbb{K}[x_0, x_1, \dots, x_{n-1}]$ over a field
$\mathbb{K}$. In this paper, we compute the $\mathbb{N}$-graded Betti numbers
of the edge ideals of three families of circulant graphs
$C_n(1,2,\dots,\widehat{j},\dots,\lfloor \frac{n}{2} \rfloor)$,
$C_{lm}(1,2,\dots,\widehat{2l},\dots, \widehat{3l},\dots,\lfloor \frac{lm}{2}
\rfloor)$ and $C_{lm}(1,2,\dots,\widehat{l},\dots,\widehat{2l},\dots,
\widehat{3l},\dots,\lfloor \frac{lm}{2} \rfloor)$. Other algebraic and
combinatorial properties like regularity, projective dimension, induced
matching number and when such graphs are well-covered, Cohen-Macaulay,
Sequentially Cohen-Macaulay, Buchsbaum and $S_2$ are also discussed.
