A degree sequence of an undirected graph 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 the fact that each edge connects two vertices and is thus counted twice (Skiena 1990, p. 157).

A sequence of nonnegative integers that is the degree sequence of some simple graph is called a graphic sequence. The Erdős-Gallai theorem gives a necessary and sufficient criterion for this property.

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 (4, 4, 4, 3, 3, 3, 3, 3, 3).

A graph determined up to graph isomorphism by its degree sequence is said to be unigraphic or called a "unigraph" (Tyshkevich 2000, Barrus et al. 2025).

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 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 k-connected graph corresponding to the degree sequence. For example, while the degree sequence is 1- but not 2-connected, is 2-connected.