There are several different definitions of the barbell graph.
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 
.
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 is 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
and the latter has recurrence equation
 |
(4) |
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."
See also
Dumbbell Curve, Flower Graph, Kayak Paddle Graph, Lollipop Graph, Tadpole Graph
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References
Ghosh, A.; Boyd, S.; and Saberi, A. "Minimizing Effective Resistance of a Graph." Proc. 17th Internat. Sympos. Math. Th. Network and Systems, Kyoto, Japan, July 24-28, 2006. pp. 1185-1196.Herbster, M. and Pontil, M. "Prediction on a Graph with a Perception." Neural Information Processing Systems Conference, 2006. http://eprints.pascal-network.org/archive/00002892/01/boundgraph.pdf.Northrup, A. "A Study of Semiregular Graphs." Senior research paper. Stetson University, 2002. http://www.stetson.edu/artsci/mathcs/students/research/math/ms498/2001/alison/finaldraft.pdf.Wilf, H. S. "The Editor's Corner: The White Screen Problem." Amer. Math. Monthly 96, 704-707, 1989.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Barbell Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BarbellGraph.html