Invariants of the symbolic powers of edge ideals

Abstract

Let G be a graph and I = I(G) be its edge ideal. When G is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of I and compute the Waldschmidt constant. When G is complete graph then we describe the generators of the symbolic powers of I and compute the Waldschmidt constant and the resurgence of I. Moreover for complete graph we prove that the Castelnuovo-Mumford regularity of the symbolic powers and ordinary powers of the edge ideal coincide.

Regularity of symbolic powers of certain graphs

“…If r = ⌊ n 2 ⌋ − 1 and n is odd then G n,r gives the class of odd cycles which has been studied by Gu et al in [8]. If r = 0 then G n,r gives the class of complete graph which we have studied in [3]. Both of the above results follow as a special case of our result.…”

“…In this direction in [8], Gu et al have proved the conjecture for odd cycles. Recently many researchers are working in this direction and in [12], Jayanthan and Kumar have proved the conjecture for certain class of unicyclic graph, in [3] we proved it for complete graph and in [6], [7], Seyed Fakhari has solved the conjecture for unicyclic graph and chordal graph respectively. In [4], DiPasquale et al have studied resurgence and asymptotic resurgence.…”

Symbolic defects of edge ideals of unicyclic graphs

“…But it is still difficult to compute the number of minimal generators of D(s) if G contains an induced odd cycle. Recently, the invariants associated with the symbolic powers of edge ideals of some simple graphs have been studied in [2,6,8]. M. Janssen et al compute the sdefect(I, s) for n + 1 ≤ s ≤ 2n + 1, where I is the edge ideal of an odd cycle C 2n+1 in [8].…”

Comparing symbolic powers of edge ideals of weighted oriented graphs

“…For any homogeneous ideal I ⊂ R, its s-th symbolic power is defined as I (s) = ⋂ p∈Ass I (I s R p ∩ R). Symbolic powers and invariants of the edge ideals of simple graphs have been studied in [7,12] and [15]. Recently, we have studied the symbolic powers and some invariant of the edge ideals of weighted oriented graphs in [16].…”