  Let $G$ be a simple graph and $I(G)$ be its edge ideal. In this article, 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 $h$-polynomial. Also, we
demonstrate that the depth of edge ideals of graphs is independent from the
regularity and degree of $h$-polynomial by constructing graphs.
