For every graph $G$ with size $m$ and no connected component isomorphic to $K_2$, we prove that, for $L=(1,1,2,2,\dots,\lfloor m/2 \rfloor+2,\lfloor m/2 \rfloor+2)$, we can assign labels of $L$ to the edges of $G$ in an injective way so that no two adjacent vertices of $G$ are incident to the same sum of labels. This implies that every such graph with size $m$ can be labelled in an equitable and proper way with labels from $\{1,\dots,\lfloor m/2 \rfloor+2\}$, which improves on a result proved by Haslegrave, and Szabo Lyngsie and Zhong, implying this can be achieved with labels from $\{1,\dots,m\}$.