A word-representable graph is a finite simple graph for which there is a finite word over its vertex set, with each graph vertex appearing as a letter such that two distinct letters alternate in the word precisely when the corresponding vertices are adjacent. A representing word is -uniform if every graph vertex occurs exactly times, where is a positive integer. The least such is the representation number .

For example, the word 14213243 represents the cycle graph with edges , , , and . The corresponding subwords obtained by retaining each edge's two letters are 1212, 1414, 2323, and 4343, which alternate. The remaining pairs give 1133 and 4224, which do not alternate. Thus this is a 2-uniform representation.

Akgün et al. (2019) enumerated the connected graphs that are not word-representable up to graph isomorphism. For , 2, ..., their numbers begin 0, 0, 0, 0, 0, 1, 25, 929, 54957, 4880093, 650856040, ... (OEIS A290814). A non-word-representable graph is minimal when every proper induced subgraph is word-representable. For , 6, ..., the numbers of minimal examples begin 0, 1, 10, 47, 179, ... (OEIS A319491).