An integral graph, not to be confused with an integral embedding of a graph, is defined as a graph whose graph spectrum consists entirely of integers. The
notion was first introduced by Harary and Schwenk (1974). The numbers of simple integral
graphs on 
,
2, ... nodes are 0, 2, 3, 6, 10, 20, 33, 71, ... (OEIS A077027),
illustrated above for small 
.
The numbers of connected simple integral graphs on 
, 2, ... nodes are 1, 1, 1, 2, 3, 6, 7, 22, 24, 83, ... (OEIS
A064731), illustrated above for small 
.
The following table lists common graph classes and the their members which are integral.
| graph | integral for 
of the form |
| antiprism graph | 3 |
complete
graph  | all |
cycle
graph  | 2, 3, 4, 6 |
| empty graph | all |
| prism graph | 3, 4, 6 |
star graph  |  |
wheel
graph  | 4 |
The following table lists some special named graphs that are integral and gives their spectra.
-simplex
