Geodesic
Complementary Pairs of Graphs With the Second Largest Eigenvalue not Exceeding (sqrt(5)-1)/2
Publications de l'Institut Mathématique, _N_S_57 (1995) no. 71, p. 179 .

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We characterize (in terms of minimal forbidden subgraphs) graphs having the following property: both the graph and its complement have the second largest eigenvalue not exceeding $(\sqrt{5}-1)/2$, i.e. the golden section. This characterization also enables us to find explicitely all graphs in question.
Classification : 05C50 
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     title = {Complementary {Pairs} of {Graphs} {With} the {Second} {Largest} {Eigenvalue} not {Exceeding} (sqrt(5)-1)/2},
     journal = {Publications de l'Institut Math\'ematique},
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     volume = {_N_S_57},
     number = {71},
     year = {1995},
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Slobodan K. Simić. Complementary Pairs of Graphs With the Second Largest Eigenvalue not Exceeding (sqrt(5)-1)/2. Publications de l'Institut Mathématique, _N_S_57 (1995) no. 71, p. 179 . http://geodesic.mathdoc.fr/item/PIM_1995_N_S_57_71_a18/