A unicyclic graph is a connected graph containing exactly one cycle (Harary 1994, p. 41). A connected 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.
Thus a connected graph that is unicyclic has the same number of edges as vertices.
More generally, a graph with exactly one cycle and 
connected components
satisfies 
.
This follows since deleting an edge from the unique 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 component creates exactly one 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 2018).
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.
