Characterizing line graphs by star complements Original Research Article

In this paper it is shown that, for any odd integer t>3, the line graph L(Kt) is the unique maximal graph having the cycle Ct 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(Kt) is the unique maximal graph having Crunion or logical sumCs as a star complement for the eigenvalue −2.