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. The Kneser graph <math>KG_K G_{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 set]]s, having no elements in common. In the special 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 disjoint from both of them, consisting of all the elements that are neither in <math>X</math> nor in <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.
