Let G be the circulant graph Cn(S) with S⊆{1,2,…,⌊n2⌋}, and let I(G) denote the edge ideal in the polynomial ring R=𝕂[x0,x1,…,xn−1] over a field 𝕂. In this paper we compute the ℕ-graded Betti numbers of the edge ideals of three families of circulant graphs Cn(1,2,…,j^,…,⌊n2⌋), Clm(1,2,…,2l^,…,3l^,…,⌊lm2⌋) and Clm(1,2,…,l^,…,2l^,…,3l^,…,⌊lm2⌋). 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 S2 are also discussed.