An isolated vertex of a graph is a graph vertex of vertex degree 0. Terminology is not
uniform: Hartsfield and Ringel (1990, p. 8), D'Angelo and West (2000, p. 212),
and West (2000, p. 22) use isolated vertex , while Harary (1994, p. 15)
uses isolated point .
The numbers of unlabeled simple graphs with
no isolated vertices are 0, 1, 2, 7, 23, 122, 888, ... for A002494 ),
the first few of which are illustrated above. Each such graph
has a graphical partition of length graphs can have
the same degree sequence . For example, a five-vertex
path graph and the graph
disjoint union of a triangle graph and a two-vertex
complete graph both have degree
sequence
A connected graph with more than one vertex has
no isolated vertices.
See also Connected Graph ,
Endpoint ,
Graph Vertex ,
Graphical
Partition ,
Vertex Degree
Explore with Wolfram|Alpha
References D'Angelo, J. P. and West, D. B. Mathematical Thinking: Problem-Solving and Proofs, 2nd ed. Harary, F. Graph
Theory. Hartsfield, N. and
Ringel, G. Pearls
in Graph Theory: A Comprehensive Introduction. Sloane, N. J. A. Sequence A002494 /M1762
in "The On-Line Encyclopedia of Integer Sequences." West, D. B.
Introduction
to Graph Theory, 2nd ed. Referenced
on Wolfram|Alpha Isolated Vertex
Cite this as:
Weisstein, Eric W. "Isolated Vertex."
From MathWorld https://mathworld.wolfram.com/IsolatedVertex.html
Subject classifications
-vertex simple graphs with
no isolated vertices are 0, 1, 2, 7, 23, 122, 888, ... for 
, 2, ... (OEIS A002494),
the first few of which are illustrated above. Each such graph
has a graphical partition of length 
, but nonisomorphic graphs can have
the same degree sequence. For example, a five-vertex
path graph and the graph
disjoint union of a triangle graph and a two-vertex
complete graph both have degree
sequence 
.
