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. 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

			(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."