I think this is false and there are counterexamples.
Here is a sage session.
sage: ed,e=([(0, 2), (0, 4), (0, 5), (1, 3), (1, 5), (2, 4), (3, 5)], (0, 2))
sage: G=Graph(ed);F=Graph(G);F.delete_edge(e)
sage: eg=G.adjacency_matrix().eigenvalues();eg.sort(reverse=1);eg
[2.414213562373095?, 1.732050807568878?, -0.4142135623730951?, -1, -1, -1.732050807568878?]
sage: ef=F.adjacency_matrix().eigenvalues();ef.sort(reverse=1);ef
[2.228328215946636?, 1.360409337131395?, 0.1858853464264596?, -1, -1, -1.774622899504490?]