In the mathematical field of [[graph theory]], a '''word-representable graph''' is a [[graph (discrete mathematics)|graph]] is '''word-representable''' if itthat can be characterisedcharacterized by a word (or sequence) whose entries alternate in a prescribed way. In particular, if the vertex set of the graph is ''V'', one should be able to choose a word ''w'' over the alphabet ''V'' such that letters ''a'' and ''b'' alternate in ''w'' if and only if the pair ''ab'' is an edge in the graph. (Letters ''a'' and ''b'' '''alternate''' in ''w'' if, after removing from ''w'' all letters but the copies of ''a'' and ''b'', one obtains a word ''abab''... or a word ''baba''....) For example, the [[cycle graph]] labeled by ''a'', ''b'', ''c'' and ''d'' in clock-wise direction is word-representable because it can be represented by ''abdacdbc'': the pairs ''ab'', ''bc'', ''cd'' and ''ad'' alternate, but the pairs ''ac'' and ''bd'' do not.
The word ''w'' is ''G''<nowiki/>'s ''word-representant'', and one says that that ''w'' ''represents'' ''G''. The smallest (by the number of vertices) non-word-representable graph is the [[wheel graph]] ''W''<sub>5</sub>, which is the only non-word-representable graph on 6 vertices.
