Symbolic Powers and Symbolic Rees Algebras of Binomial Edge Ideals of Some Classes of Block Graphs

TL;DR

This paper studies the properties of symbolic powers and Rees algebras of binomial edge ideals in specific block graphs, revealing conditions under which these ideals coincide or exhibit regularity properties.

Contribution

It establishes new results on symbolic powers, $F$-splitness, and $F$-regularity of binomial edge ideals for certain classes of block graphs.

Findings

Symbolic powers of binomial edge ideals of pendant cliques graphs equal their ordinary powers.

Binomial edge ideals of generalized caterpillar graphs are symbolic $F$-split.

Symbolic Rees algebras of these ideals are strongly $F$-regular.

Abstract

In this paper, we investigate some properties of symbolic powers and symbolic Rees algebras of binomial edge ideals associated with some classes of block graphs. First, it is shown that symbolic powers of binomial edge ideals of pendant cliques graphs coincide with the ordinary powers. Furthermore, we see that binomial edge ideals of a generalization of these graphs are symbolic $F$-split. Consequently, net-free generalized caterpillar graphs are also a class of block graphs with symbolic $F$-split binomial edge ideals. Finally, it turns out that symbolic Rees algebras of binomial edge ideals associated with these two classes, namely pendant cliques graphs and net-free generalized caterpillar graphs, are strongly $F$-regular.