Given an undirected graph, a degree sequence is a monotonic nonincreasing sequence of the vertex degrees (valencies) of its graph vertices. The number of degree sequences for a graph of a given order is closely related to graphical partitions. The sum of the elements of a degree sequence of a graph is always even due to fact that each edge connects two vertices and is thus counted twice (Skiena 1990, p. 157).
The minimum vertex degree in a graph 
is denoted 
, and the maximum
vertex degree is denoted 
(Skiena 1990, p. 157). A graph
whose degree sequence contains only multiple copies of a single integer is called
a regular graph. A graph corresponding to a given
degree sequence 
can be constructed in the Wolfram Language
using RandomGraph[DegreeGraphDistribution[d]].
It is possible for two topologically distinct graphs to have the same degree sequence. Moreover, two distinct convex polyhedra can even have the same degree sequence for their skeletons, as exemplified by the triangular cupola and tridiminished icosahedron Johnson solids, both of which have 8 faces, 9 vertices, 15 edges, and degree sequence (3, 3, 3, 3, 3, 3, 4, 4, 4).
A graph having a unique degree sequence may be said to be unigraphic or called a "unigraph" (Tyshkevich 2000, Barrus et al. 2023).
The number of distinct degree sequences for graphs of 
, 2, ... nodes are given by 1, 2, 4, 11, 31, 102, 342, 1213,
4361, ... (OEIS A004251), compared with the
total number of nonisomorphic simple undirected graphs with 
graph vertices of 1, 2, 4,
11, 34, 156, 1044, ... (OEIS A000088). The
first order having fewer degree sequences than number of nonisomorphic graphs is
therefore 
.
For the graphs illustrated above, the degree sequences are given in the following
table.
| 1 |  |
| 2 |  ,  |
| 3 |  ,  ,  ,  |
| 4 |  ,  ,  ,  , |
 ,  ,  ,  , | |
 ,  ,  |
The possible sums of elements for a degree sequence of order 
are 0, 2, 4, 6, ..., 
.
A degree sequence is said to be 
-connected if there exists some 
-connected graph corresponding
to the degree sequence. For example, while the degree sequence 
is 1- but not 2-connected, 
is 2-connected.
