Abstract
We compute the Castelnuovo–Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to combine bounds on the regularity, the projective dimension, and the reduced Euler characteristic to derive an exact value for the regularity.
Highlights
Let G be any finite simple graph with vertex set V ( G ) = { x1, . . . , xn } and edge set E( G ), where simple means no loops or multiple edges
Relating the homological invariants of I ( G ) and the graph theoretic invariants of G has proven to be a fruitful approach to building this dictionary
We denote by β i,j ( I ( G )) the i, jth graded Betti number of I ( G ); this number equals the number of minimal generators of degree j in the ith syzygy module of I ( G )
Summary
The value of reg( I (Cn )) can be deduced from the work of Jacques ([13], Theorem 7.6.28) One can view these circulant graphs as “extremal” cases in the sense that |S| is either as large or as small as possible. To generalize the case of Cn (a circulant graph where every vertex has degree two), we compute the regularity of the edge ideal of any cubic (every vertex has degree three) circulant graph, that is G = C2n ( a, n) with 1 ≤ a ≤ n We first recall the relevant background regarding graph theory and commutative algebra, along with our new result on the regularity of square-free monomial ideals.
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