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 3barbell 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 2path graph "barbell graphs." This version might perhaps be better called a "double flower graph."