A connection between the bipartite complements of line graphs and the line graphs with two positive eigenvalues

In 1974 Cvetković and Simić showed which graphs $G$ are the bipartite complements of line graphs. In 2002 Borovićanin showed which line graphs $L(H)$ have third largest eigenvalue $λ_3\leq0$. Our first observation is that two of the graphs Borovićanin found are the complements of two of the graphs found by Cvetković and Simić. Using the Courant-Weyl inequalities we show why this is and reprove the result of Borovićanin, highlighting some features of the graphs found by both.

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