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A proper edge $k$-colouring of a graph $G=(V,E)$ is an assignment $c:E\to\{1,2,\ldots,k\}$ of colours to the edges of the graph such that no twoadjacent edges are associated with the same colour. A neighbour sumdistinguishing edge $k$-colouring, or nsd $k$-colouring for short, is a properedge $k$-colouring such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ forevery edge $uv$ of $G$. We denote by $\chi'_{\sum}(G)$ the neighbour sumdistinguishing index of $G$, which is the least integer $k$ such that an nsd$k$-colouring of $G$ exists. By definition at least maximum degree, $\Delta(G)$colours are needed for this goal. In this paper we prove that $\chi'_\Sigma(G)\leq \Delta(G)+1$ for any graph $G$ without isolated edges and with ${\rmmad}(G)<3$, $\Delta(G) \geq 6$.

On the neighbour sum distinguishing index of graphs with bounded maximum average degree