Algebraic Properties of Product of Graphs

Abstract

Abstract. Let G and H be two simple graphs and let G * H denotes the graph theoretical product of G by H. In this paper we provide some results on graded Betti numbers, Castelnuovo-Mumford regularity, projective dimension, h-vector, and Hilbert series of G * H in terms of that information of G and H. To do this, we will provide explicit formulae to compute graded Betti numbers, h-vector, and Hilbert series of disjoint union of complexes. Also we will prove that the family of graphs whose regularity equal the maximum number of pairwise 3-disjoint edges, is closed under product of graphs.

Graded Betti numbers of some circulant graphs

“…However, this formula is somewhat daunting to use for computing all Betti numbers of circulant graphs with large number of vertices because one has to compute the dimensions of all the homology groups. In this paper, we see the circulant graphs from different lens and collectively use [15], Hochster's formula and [13], Corollary 3.4 to compute the Betti numbers and the regularity.…”

Betti numbers of some circulant graphs

“…In this section we provide explicit formulae for computing the Betti numbers in linear strand of the circulant graph C 2n (O(n) ∪ {2}). To do this, in view of Lemma 3.1, we make use of two ingredients: the first is the formula for Betti numbers of the product of graphs given in [15], Corollary 3.4 or [23], Lemma 5.4, and the second is the formula for Betti numbers of cycles described in [11], Sections 7.4 and 7.5. We first recall them.…”

“…Using the formula for Betti numbers of the product of graphs (see [15], Corollary 3.4 or [23], Lemma 5.4) together with the Betti numbers of cycles described in [11], Sections 7.4 and 7.5 we will provide explicit formulae for computing N-graded Betti numbers of the circulant graph C 2n (O(n) ∪ {2}). Indeed, in view of [15], Proposition 3.12 one has reg(C 2n (O(n) ∪ {2})) = reg(C n ). Hence we will prove the following.…”

“…[15], Corollary 3.4 or[23],Lemma 5.4). Let G and H be two simple graphs with disjoint vertex sets having m and n vertices, respectively.…”

Certain algebraic invariants of edge ideals of join of graphs

“…Let G and H be graphs with reg(I(G)) = 3 and reg(I(H)) ≤ 3. Then, by[13, Proposition 3.12], reg(I(G * H)) = 3. For example, if G c is triangle free which is not a forest and H is a bipartite graph whose edge ideal has regularity 3 ([4, Theorem 4.1]) or co-chordal graph, then by Corollary 4.6, reg(I(G * H)) = 3.…”