singlevertex

Singlevertex, usually called the single-vertex graph, is the graph consisting of a single vertex and no edges. In standard graph-theoretic notation it can be described as G = (V, E) with V = {v} and E = ∅. It is commonly denoted K1, the complete graph on one vertex, though some sources reserve K1 for this precise case.

Basic properties of the single-vertex graph are straightforward. Its order is 1 and its size is 0. The graph is connected by definition, since there is a trivial path from the vertex to itself of length zero. The degree of the sole vertex is 0. Its automorphism group is trivial, reflecting that there is only one vertex to map to itself. Invariants such as the chromatic number and clique number are both 1, and the independence number is also 1.

Role and uses in graph theory are mainly as a fundamental base case and a building block. It appears in inductive proofs as the simplest non-empty graph, and it serves as a minimal example when illustrating concepts such as connectivity, degree, and simple graph invariants. In graph constructions, adjoining a single vertex to a graph via disjoint union adds an isolated vertex, while various product and composition operations yield larger graphs starting from K1.

Related concepts include the trivial graph (the empty graph, with no vertices), the complete graphs K_n for n ≥ 2, and isolated vertices in larger graphs. See also K1, trivial graph, isolated vertex, and basic graph invariants.