Cohen–Macaulay Circulant Graphs

Abstract

Abstract. Let G be the circulant graph C n (S) with S ⊆ {1, 2, . . . , ⌊ n 2 ⌋}, and let I(G) denote its the edge ideal in the ring R = k[x 1 , . . . , x n ]. We consider the problem of determining when G is Cohen-Macaulay, i.e, R/I(G) is a Cohen-Macaulay ring. Because a Cohen-Macaulay graph G must be well-covered, we focus on known families of wellcovered circulant graphs of the form C n (1, 2, . . . , d). We also characterize which cubic circulant graphs are Cohen-Macaulay. We end with the observation that even though the well-covered property is preserved under lexicographical products of graphs, this is not true of the Cohen-Macaulay property.

The Regularity of Some Families of Circulant Graphs

“…In this note we give explicit formulas for reg(I(G)) for the edge ideals of two infinite families of circulant graphs. Our results complement previous work on the algebraic and combinatorial topological properties of circulant graphs (e.g, [4,12,14,15,16,17,18]). Fix an integer n ≥ 1 and a subset S ⊆ {1, .…”

On the reduced Euler characteristic of independence complexes of circulant graphs

“…We focused on the property of n, but nice combinatorial properties like well-coveredness (see [7], [5]), strongly connectedness (see [10]), vertex decomposability and shellability (see [12]) could be helpful. From another point of view, it would be nice to find entire classes of circulants that for particular n and S have vanishing Euler characteristic, by using a theoretical approach rather than the computational one used in Example 2.10.…”

Independence Complexes of Well-Covered Circulant Graphs

“…Theorem 3.3 shows that the independence complexes of these graphs are always Buchsbaum. In addition, we classify when these complexes are vertex decomposable, solving an open problem of [29]. We also include a discussion on the h-vectors of Buchsbaum complexes.…”

“…Finally, we would like to add a small erratum to [29]. On page 1902, the f -vector and hvector on line 4 should be (1,11,33,22), respectively, (1,8,14 [17] contained a proof for the equivalence of Theorem 3.4 (ii) and (iv) of [29].…”