A graph blow-up of a graph 
with vertices 
, 
,
..., 
is a graph obtained by replacing each vertex 
by a nonempty independent
vertex set 
and replacing each graph edge 
by all the edges of the
complete bipartite graph with parts 
and 
, with no additional edges (Illingworth 2022). Thus all vertices in a given set 
have the same graph neighborhood
outside that set. The natural map from the blow-up to 
sends every vertex
of 
to 
, so 
is the fiber over 
.
If 
, the resulting graph is often
denoted ![H[m_1,m_2,...,m_p]](/images/equations/GraphBlow-Up/Inline17.svg)
.
Its graph order and edge
count are
![|V(H[m_1,m_2,...,m_p])|=sum_(i=1)^pm_i,](/images/equations/GraphBlow-Up/NumberedEquation1.svg) |
and
![|E(H[m_1,m_2,...,m_p])|=sum_(v_iv_j in E(H))m_im_j.](/images/equations/GraphBlow-Up/NumberedEquation2.svg) |
When all 
,
the blow-up is called uniform and is denoted 
by some authors. This exact-equality case is also called
a balanced blow-up (Hatami et al. 2014). It is the graph
lexicographic product 
, where 
is the empty graph on 
vertices.
When the graph order is fixed but is not necessarily
divisible by 
,
"balanced" is also used when the fiber sizes
differ by at most one (Avigad and Goldreich 2011).
Blow-ups of complete graphs are precisely the complete multipartite graphs. In particular, balanced blow-ups of complete graphs are Turán graphs.
This graph operation is distinct from algebraic blow-up and finite-time blow-up, and from the blow-up lemma, which concerns embedding graphs into regular pairs that approximate pairs of vertex sets inducing complete bipartite graphs in a graph blow-up.
