Symbolic powers of edge ideals of graphs

Abstract

Let G be a graph and let I = I(G) be its edge ideal. When G is unicyclic, we give a decomposition of symbolic powers of I in terms of its ordinary powers. This allows us to explicitly compute the Waldschmidt constant and the resurgence number of I. When G is an odd cycle, we explicitly compute the regularity of I (s) for all s ∈ N. In doing so, we also give a natural lower bound for the regularity function reg I (s) , for s ∈ N, for an arbitrary graph G.2010 Mathematics Subject Classification. 13D02, 13P20, 13F55.

Invariants of the symbolic powers of edge ideals

“…They have described the generators of the symbolic powers of the edge ideal of an odd cycle by using the concept of minimal vertex cover and calculated the invariants associated to the symbolic powers of the edge ideal of the same graph. In [5] Gu et al have extended these results for the unicyclic graph by explicitly computing the generators of the symbolic powers. Another important invariant in commutative algebra is Castelnuovo-Mumford regularity.…”

“…It has been conjectured by N. C. Minh that for a finite simple graph G reg I(G) (s) = reg I(G) s for s ∈ N. The conjecture is true for bipartite graph. In [5] Gu et al have proved the conjecture for odd cycles. Recently in [10] Jayanthan and Kumar have proved the conjecture for certain class of unicyclic graph and in [13] Seyed Fakhari has solved the conjecture for unicyclic graph.…”

“…The work of this paper is mainly motivated by the papers [9] and [5]. In this paper we study the structure of the symbolic powers of edge ideal of clique sum of two different odd length cycles joined at a single vertex and prove Minh's conjecture for the class of complete graphs.…”

Regularity of symbolic powers of edge ideals of unicyclic graphs

“…It is also reasonable to study the regularity of symbolic powers of edge ideals. In this direction, Minh posed the following conjecture (see [15]). In the above conjecture, I(G) (s) , denotes the s-th symbolic powers of I(G).…”

“…Therefore, Conjecture 1.2 is trivially true for this class of graphs. On the other, it is known that Conjecture 1.2 is also true for cycles, unicyclic graphs and Cameron-Walker graphs (see [15,29,31], respectively). Moreover, Jayanthan and Kumar [20] proved Conjecture 1.2 for some classes of graphs which are obtained by the clique sum of odd cycles and bipartite graphs.…”