Let $\gamma'_s(G)$ be the signed edge domination number of G. In 2006, Xu
conjectured that: for any $2$-connected graph G of order $ n (n \geq 2),$
$\gamma'_s(G)\geq 1$. In this article we show that this conjecture is not true.
More precisely, we show that for any positive integer $m$, there exists an
$m$-connected graph $G$ such that $ \gamma'_s(G)\leq -\frac{m}{6}|V(G)|.$ Also
for every two natural numbers $m$ and $n$, we determine $\gamma'_s(K_{m,n})$,
where $K_{m,n}$ is the complete bipartite graph with part sizes $m$ and $n$.