Barbell Graph

There are several different definitions of the barbell graph.


Most commonly and in this work, the n-barbell graph is the simple graph obtained by connecting two copies of a complete graph K_n by a bridge (Ghosh et al. 2006, Herbster and Pontil 2006). The 3-barbell graph is isomorphic to the kayak paddle graph KP(3,3,1).

Precomputed properties of barbell graphs are available in the Wolfram Language as GraphData[{"Barbell", n}].

Barbell graphs are geodetic.

By definition, the n-barbell graph has cycle polynomial is given by


where C_(K_n)(x) is the cycle polynomial of the complete graph K_n. Its graph circumference is therefore n.

The n-barbell graph has chromatic polynomial and independence polynomial


and the latter has recurrence equation


Wilf (1989) adopts the alternate barbell convention by defining the n-barbell graph to consist of two copies of K_n connected by an n-path.

Northrup (2002) calls the graphs obtained by joining n 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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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., A. "A Study of Semiregular Graphs." Senior research paper. Stetson University, 2002., H. S. "The Editor's Corner: The White Screen Problem." Amer. Math. Monthly 96, 704-707, 1989.

Referenced on Wolfram|Alpha

Barbell Graph

Cite this as:

Weisstein, Eric W. "Barbell Graph." From MathWorld--A Wolfram Web Resource.

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