Certain [[Kneser graph]]s, graphs constructed from the intersection patterns of equal-size sets, are locally linear. Kneser graphs are described by two parameters, the size of the sets they represent and the size of the universe from which these sets are drawn:. theThe Kneser graph <math>KG_{a,b}</math> has <math>\tbinom{a}{b}</math> vertices (in the standard notation for [[binomial coefficient]]s), representing the <math>b</math>-element subsets of an <math>a</math>-element set. In this graph, two vertices are adjacent when the corresponding subsets are [[disjoint.Forset]]s, parametershaving thatno obeyelements in common. In the equationspecial case when <math>a=3b</math>, the resulting graph is locally linear, because for each two disjoint <math>b</math>-element subsets <math>X</math> and <math>Y</math> there is exactly one other <math>b</math>-element subset (thedisjoint complementfrom both of theirthem, union)consisting of all the elements that isare disjointneither fromin both<math>X</math> ofnor themin <math>Y</math>. The resulting locally linear graph has <math>\tbinom{3b}{b}</math> vertices and <math>\tfrac{1}{2}\tbinom{3b}{b}\tbinom{2b}{b}</math> edges. For instance, for <math>b=2</math> the Kneser graph <math>KG_{6,2}</math> is locally linear with 15 vertices and 45 edges.{{r|lpv}}
Locally linear graphs can also be constructed from progression-free sets of numbers.
