Certain algebraic invariants of edge ideals of join of graphs

Abstract

Let [Formula: see text] be a simple graph and [Formula: see text] be its edge ideal. In this paper, we study the Castelnuovo–Mumford regularity of symbolic powers of edge ideals of join of graphs. As a consequence, we prove Minh’s conjecture for wheel graphs, complete multipartite graphs, and a subclass of co-chordal graphs. We obtain a class of graphs whose edge ideals have regularity three. By constructing graphs, we prove that the multiplicity of edge ideals of graphs is independent from the depth, dimension, regularity, and degree of [Formula: see text]-polynomial. Also, we demonstrate that the depth of edge ideals of graphs is independent from the regularity and degree of [Formula: see text]-polynomial by constructing graphs.