Abstract:  Suppose $f$ is a one-to-one mapping from $V(G)$ onto $\{1,2,\cdots,|V(G)|\}$. Let $BS(G,f)=\sum\limits_{uv\in E(G)}|f(u)-f(v)|$. The bandwidth sum of $G$, denoted by $BS(G)$, is $BS(G)=\min\limits_{f}BS(G,f)$. In this paper, we obtain the relationship between $BS(G+e)$ and $BS(G)$, where $e\in\overline{E(G)}$, $BS(G)+1\leq BS(G+e)\leq BS(G)+n-1$. We also show that these bounds are sharp.

Key words: graph, labelling, bandwidth sum