Abstract
In this paper it is shown that, for any odd integer t>3, the line graph L( K t ) is the unique maximal graph having the cycle C t as a star complement for the eigenvalue −2. This result yields a characterization of L( G) for Hamiltonian graphs G with an odd number of vertices. We also show that, if t= r+ s, where r and s are odd integers >1, then, provided that t≠8, L( K t ) is the unique maximal graph having C r ∪ C s as a star complement for the eigenvalue −2.
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