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

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.