A unicyclic graph is a connected graph containing exactly one graph cycle (Harary 1994, p. 41). A unicyclic graph is therefore a pseudotree that is not a tree. Equivalently, a connected graph is unicyclic iff its circuit rank, also called its cyclomatic number, is 1. Consequently, a graph is a pseudoforest iff each of its connected components is either a tree or a unicyclic graph.
Thus a connected graph that is unicyclic has the same number of edges as vertices.
More generally, a graph with exactly one graph cycle and 
connected components satisfies 
.
This follows since deleting an edge from the unique graph cycle leaves a forest
with the same number 
of connected components and hence 
edges. Conversely, adding
an edge to a forest between
two vertices in the same connected
component creates exactly one graph cycle. In
particular, attaching pendant edges, or more generally
trees, to vertices of a
unicyclic graph preserves unicyclicity.
Truszczyński (1984) conjectured that all unicyclic graphs except the cycle graph 
with 
or 2 (mod 4) are graceful (Gallian 2025).
The numbers of unicyclic graphs on 
, 2, ... vertices are 0,
0, 1, 3, 9, 25, 68, 185, ... (OEIS A236570),
the first few of which are illustrated above.
The corresponding numbers of connected unicyclic graphs are 0, 0, 1, 2, 5, 13, 33, 89, ... (OEIS A001429), the first few of which are illustrated above.
Examples of unicyclic classes of graphs include 
-caveman graphs, cycle
graphs 
,
pan graphs, sunlet graphs 
, and tadpole
graphs.
