  In the present paper, we study the upper and lower bounds for the number of
facets of symmetric edge polytopes of connected graphs conjectured by Braun and
Bruegge. In particular, we show that their conjecture is true for any graph
that is the join of two graphs (equivalently, for any connected graph whose
complement graph is not connected). It is known that any symmetric edge
polytope is a centrally symmetric reflexive polytope. Hence our results give a
partial answer to Nill's conjecture: the number of facets of a $d$-dimensional
reflexive polytope is at most $6^{d/2}$.
