A graph can be determined to be nonhamiltonian in the Wolfram Language using HamiltonianGraphQ[g]
and precomputed values are available for a number of named graphs using GraphData[graph,
"Nonhamiltonian"].

The numbers of connected simple nonhamiltonian graphs on , 2, ... nodes are 0, 1, 1, 3, 13, 64, 470, 4921, ... (OEIS
A126149), the first few of which are illustrated
above, and the corresponding number of not-necessarily-connected simple nonhamiltonian
graphs on ,
2, ... nodes are 0, 2, 3, 8, 26, 108, 661, 6150, 97585, ... (OEIS A246446).

There are no nonhamiltonian polyhedral graphs on 10 and fewer nodes, and the numbers for , 12, ... are 74, 1600, 43984, 1032208, 22960220, ... (OEIS
A007033). The most notable of 11-node graphs
are the Herschel graph and Goldner-Harary
graph.

The following table summarizes some small named connected nonhamiltonian graphs.

Aldred, R. E. L.; Bau, S.; Holton, D. A.; and McKay, B. D. "Nonhamiltonian 3-Connected Cubic Planar Graphs."
SIAM J. Disc. Math.13, 25-32, 2000.Royle, G. "Cubic
Symmetric Graphs (The Foster Census): Hamiltonian Cycles." http://school.maths.uwa.edu.au/~gordon/remote/foster/#hamilton.Sloane,
N. J. A. Sequences A007033/M5351,
A126149, and A246446
in "The On-Line Encyclopedia of Integer Sequences."