The assembly number 
of a finite connected simple graph 
with vertex set 
is the number of its assembly trees using the edge gluing
rule (Vince and Bóna 2012, p. 2). An assembly tree is a finite rooted tree whose nodes
are labeled by subsets of 
. Its leaves have the singleton
sets 
as labels, one for each 
,
and its root vertex is labeled by 
. Each internal node has exactly
two unordered children. Their labels are disjoint
sets whose union is that node's
label, and an edge of 
joins a vertex in one child
label to a vertex in the other. Trees
are counted up to label-preserving graph isomorphism.
For 
,
an assembly tree has 
leaves and 
internal nodes. When 
, its sole node
is treated as a leaf in this assembly-tree convention. These trees
describe how vertex-induced subgraphs
join to form 
,
a process motivated by the self-assembly of macromolecules in virus capsids (Kainen
2025, p. 220).
The assembly numbers for several parametrized graphs are summarized in the table below (Vince and Bóna 2012, pp. 4-7). The cycle graph formula applies for 
, and the other rows apply for 
. Here, 
is a binomial coefficient,
is a factorial,
and 
is a double factorial.
| family | OEIS | assembly number |
complete
bipartite graph  | A217523 | |
complete graph  | A001147 |  |
complete tripartite
graph  | A361072 | |
cycle graph  | A001700 |  |
path graph  | A000108 |  |
star graph  | A000142 |  |
If 
is a Hamiltonian graph on 
vertices, then
 |
where 
denotes the cycle graph on 
vertices. Similarly, if 
is a nonhamiltonian
graph on 
vertices with a Hamiltonian
path, then
 |
where 
denotes the path graph on 
vertices.
