Graded Betti numbers of some families of circulant graphs

Abstract

Let G be the circulant graph C n ( S ) with S ⊆ { 1 , 2 , … , ⌊ n 2 ⌋ } , 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 , … , ˆ j , … , ⌊ n 2 ⌋ ) , C l m ( 1 , 2 , … , ˆ 2 l , … , ˆ 3 l , … , ⌊ l m 2 ⌋ ) and C l m ( 1 , 2 , … , ˆ l , … , ˆ 2 l , … , ˆ 3 l , … , ⌊ l m 2 ⌋ ) . 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.