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A proper edge k-colouring of a graph $$G=(V,E)$$ is an assignment $$c:E\rightarrow \{1,2,\ldots ,k\}$$ of colours to the edges of the graph such that no two adjacent edges are associated with the same colour. A neighbour sum distinguishing edge k-colouring, or nsd k-colouring for short, is a proper edge k-colouring such that $$\sum _{e\ni u}c(e)\ne \sum _{e\ni v}c(e)$$ for every edge uv of G. We denote by $$\chi '_{\Sigma }(G)$$ the neighbour sum distinguishing 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 $$\chi '_\Sigma (G) \le \Delta (G)+1$$ for any graph G without isolated edges, with $$\mathrm{mad}(G)<3$$ and $$\Delta (G) \ge 6$$ . [ABSTRACT FROM AUTHOR]

A graph G is prime if the vertices can be distinctly labeled with the integers 1,2,...,|V(G)| so that adjacent vertices have relatively prime labels. We show that every cubic bipartite graph is prime except K3,3, which implies a number of other results. We also provide evidence to support a conjectured classification for the primality of 2-regular graphs. [ABSTRACT FROM AUTHOR]