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We adapt the method provided in an answer to Ask Sage question 54878 and define a function that provides an iterator over graphs on n vertices with the requested property.

We then treat the case of n = 6 as an example, and plot all the corresponding graphs.

Function:

def mygraphs(n):
    return (g for g in graphs.nauty_geng(f"{n} -c")
            if all(e and (1/e in ee or -1/e in ee)
                   for ee in [g.adjacency_matrix().eigenvalues()]
                   for e in ee))

List of graphs on 6 vertices with the requested property

sage: G6 = list(mygraphs(6))
sage: len(G6)
6

View them:

sage: opt = {'axes': False, 'aspect_ratio': 1, 'color': 'grey', 'alpha': 0}
sage: p = point2d([(-1.2, -1.2), (1.2, 1.2)], **opt)
sage: gg = [p + g.plot(layout='circular') for g in G6]
sage: graphics_array(gg, ncols=3)

We adapt the method provided in an answer to Ask Sage question 54878 and define a function that provides an iterator over graphs on n vertices with the requested property.

We then treat the case of n = 6 as an example, and plot all the corresponding graphs.

Function:

def mygraphs(n):
    return (g for g in graphs.nauty_geng(f"{n} -c")
            if all(e and (1/e in ee or -1/e in ee)
                   for ee in [g.adjacency_matrix().eigenvalues()]
                   for e in ee))

List of graphs on 6 vertices with the requested property

sage: G6 = list(mygraphs(6))
sage: len(G6)
6

View them:

sage: opt = {'axes': False, 'aspect_ratio': 1, 'color': 'grey', 'alpha': 0}
sage: p = point2d([(-1.2, -1.2), (1.2, 1.2)], **opt)
sage: gg = [p + g.plot(layout='circular') for g in G6]
sage: graphics_array(gg, ncols=3)