The drum graph is the 14-vertex, 21-edge cubic Hamiltonian graph illustrated above. It has girth 5, graph crossing number 2, graph diameter 4, and chromatic number 3. It is also an apex graph.
The name appears to refer to the drum-like visual appearance of the graph in the embedding presented by Robertson et al. (2019, Fig. 12).
A cubic graph is cyclically 5-connected if it has girth at least 5 and every edge cut separating two subgraphs that contain cycles has at least five edges. It is dodecahedrally-connected if no 5-edge cut having at least seven vertices on each side cuts off a subgraph that can be drawn in a disk with its five attachment vertices on the boundary.
A graph is called arched if deleting one of its edges makes it planar. The characterization of arched dodecahedrally-connected graphs states that, apart from the triplex graph itself, such a graph is arched if and only if it contains neither the Petersen graph nor the drum graph as a topological minor (Robertson et al. 2019).
The drum graph is implemented in the Wolfram Language as GraphData["DrumGraph"].
