  We prove two recent conjectures on some upper bounds for the
Castelnuovo-Mumford regularity of the binomial edge ideals of some different
classes of graphs. We prove the conjecture of Matsuda and Murai for graphs
which has a cut edge or a simplicial vertex, and hence for chordal graphs. We
determine the regularity of the binomial edge ideal of the join of graphs in
terms of the regularity of the original graphs, and consequently prove the
conjecture of Matsuda and Murai for such a graph, and hence for complete
$t$-partite graphs. We also generalize some results of Schenzel and Zafar about
complete $t$-partite graphs. We also prove the conjecture due to the authors
for a class of chordal graphs.
