a square based pyramid has 54 vertices.
a square based pyramid has 54 vertices.
how many vertices does a square based pyramid have?
54
the formula_64-regular locally linear graphs must have at least formula_65 vertices, because there are this many vertices among any triangle and its neighbors alone. (no two vertices of the triangle can share a neighbor without violating local linearity.) regular graphs with exactly this many vertices are possible only when formula_66 is 1, 2, 3, or 5, and are uniquely defined for each of these four cases. the four regular graphs meeting this bound on the number of vertices are the 3-vertex 2-regular triangle formula_67, the 9-vertex 4-regular paley graph, the 15-vertex 6-regular kneser graph formula_34, and the 27-vertex 10-regular [[complement graph]] of the [[schläfli graph]]. the final 27-vertex 10-regular graph also represents the [[intersection graph]] of the 27 lines on a [[cubic surface]].
locally linear graph
start with any vertex "v". every one of the infinitely many vertices of "g" can be reached from "v" with a simple path, and each such path must start with one of the finitely many vertices adjacent to "v". there must be one of those adjacent vertices through which infinitely many vertices can be reached without going through "v". if there were not, then the entire graph would be the union of finitely many finite sets, and thus finite, contradicting the assumption that the graph is infinite. we may thus pick one of these vertices and call it "v".
kőnig's lemma