On The Signed Edge Domination Number of Graphs

Abstract: Let $\gamma&#39;<em>s(G)$ be the signed edge domination number of G. In 2006, Xu conjectured that: for any $2$-connected graph G of order $n (n \geq 2),$ $\gamma&#39;_s(G)\geq 1$. 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&#39;_s(G)\leq -\frac{m}{6}|V(G)|.$ Also for every two natural numbers $m$ and $n$, we determine $\gamma&#39;_s(K</em>{m,n})$, where $K_{m,n}$ is the complete bipartite graph with part sizes $m$ and $n$.