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# Barbell Graph

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.

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

 (2) (3)

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."

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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.

Barbell Graph

## Cite this as:

Weisstein, Eric W. "Barbell Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BarbellGraph.html