GRADED BETTI NUMBERS OF SOME FAMILIES OF CIRCULANT GRAPHS

Let G be the circulant graph C-n (S) with S subset of {1, 2, ..., left perpendicularn/2right perpendicular}, and let I (G) denote the edge ideal in the polynomial ring R = K[x(0), x(1), ..., x(n-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 C-n (1, 2, ..., (j) over cap, ..., left perpendicularn/2right perpendicular), C-lm (1, 2, ..., (2l) over cap, ..., (3l) over cap, ..., left perpendicularlm/2 right perpendicular) and C-lm (1, 2, ..., (l) over cap , ..., (2l) over cap, ..., (3l) over cap, ..., left perpendicularlm/2right perpendicular). 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.