Objects, Invariants and Properties for Graph Theory (GT) automated conjecturing: in particular with the Sage program CONJECTURING: http://nvcleemp.github.io/conjecturing/
from:
Dutton, Ronald D., and Robert C. Brigham. "On the size of graphs of a given bandwidth." Discrete mathematics 76, no. 3 (1989): 191-195.
The definition is:
(1) a numbering is a bijection f of the vertices with 1...n (where n is the order),
(2) an edge value for edge uv with respect to f is |f(u)-f(v)|
(3) Bf is the maximum of the edge values with respect to f
(4) the bandwidth of the graph is the minimum of Bf for all bijections f.
from: Dutton, Ronald D., and Robert C. Brigham. "On the size of graphs of a given bandwidth." Discrete mathematics 76, no. 3 (1989): 191-195.
The definition is: (1) a numbering is a bijection f of the vertices with 1...n (where n is the order), (2) an edge value for edge uv with respect to f is |f(u)-f(v)| (3) Bf is the maximum of the edge values with respect to f (4) the bandwidth of the graph is the minimum of Bf for all bijections f.
So this is obviously exponential - but appears in bounds in the cited (and attached) paper. brigham-dutton-size-of-graphs-of-given-bandwidth-1989.pdf