Threevertex

Threevertex is a term encountered in several disciplines, generally denoting a vertex tied to three elements or forming part of a triadic structure. It is not a single standardized definition; its precise meaning depends on context.

In graph theory, a three-vertex (also called a tri-valent or cubic vertex) is a vertex of degree three, connected to exactly three edges. Graphs in which every vertex is a three-vertex are cubic or 3-regular graphs. Such graphs satisfy m = 3n/2 for n vertices, implying n must be even. Cubic graphs appear in chemistry as models of carbon frameworks and in network design. The tetrahedral graph, corresponding to the edges of a tetrahedron, is a classic example with four vertices each of degree three.

In computational geometry and computer graphics, three vertices determine a triangle, the simplest polygon in the plane or space. Triangular meshes rely on faces that are triangles because any three non-collinear points define a plane. Three-vertex concepts underpin mesh generation, rendering pipelines, and finite-element methods.

Usage notes: some authors may use “3-vertex” or “three-vertex” interchangeably with degree-three vertex; others reserve it for specific triadic configurations. The term’s ambiguity means precise definition should be stated in context. See also cubic graph, tetrahedron, and triangular mesh.