The term "barbell graph" has several different definitions.
Most commonly and in this work, the 
-barbell graph is the simple graph
obtained by connecting two copies of a complete graph 
by a bridge
(Ghosh et al. 2006, Herbster and Pontil 2006). The 3-barbell graph
is isomorphic to the kayak
paddle graph 
.
The 
-barbell graph is also the graph
corona product 
for 
. Each vertex
of 
together with its attached copy of
forms a complete
graph 
,
and the original edge of 
is the only edge joining the
two copies.
Precomputed properties of barbell graphs are available in the Wolfram Language as GraphData[
"Barbell", n
].
Barbell graphs are geodetic. The 
-barbell graph is ungraceful
from 
up to at least 
(E. Weisstein, Sep. 19, 2025) and likely for all larger 
.
By definition, the 
-barbell
graph has cycle polynomial given by
 |
(1)
|
where 
is the cycle polynomial of the complete
graph 
.
Its graph circumference is therefore 
.
The 
-barbell
graph has chromatic polynomial and independence
polynomial
 |  |  |
(2)
|
 |  | ![[1+(n-1)x][1+(n+1)x],](https://calendarstemplates.com/images/equations/BarbellGraph/Inline27.svg) |
(3)
|
and the latter has recurrence equation
 |
(4)
|
For 
,
the 
-barbell
graph 
has nonorientable graph genus
![gamma^~(B_n)={0 for n=3,4; 5 for n=7; 2[(n-3)(n-4)/6] otherwise,](https://calendarstemplates.com/images/equations/BarbellGraph/NumberedEquation3.svg) |
(5)
|
where ![[x]](https://calendarstemplates.com/images/equations/BarbellGraph/Inline31.svg)
is the ceiling function. This follows from the
complete graph genus formula and the theorem on
graph blocks (Ringel 1974, Stahl and Beineke 1977).
The exceptional case uses two copies of 
, each of Euler genus 2, and
one additional cross-cap.
Wilf (1989) adopts the alternate barbell convention by defining the 
-barbell graph to consist of two copies of 
connected by an 
-path.
Northrup (2002) calls the graphs obtained by joining 
bridges
on either side of a 2-path graph "barbell graphs."
This version might perhaps be better called a "double flower graph."
