A connection between the bipartite complements of line graphs and the line graphs with two positive eigenvalues
Abstract
In 1974 Cvetkovi\'c and Simi\'c showed which graphs G are the bipartite complements of line graphs. In 2002 Borovi\'canin showed which line graphs L(H) have third largest eigenvalue λ3≤0. Our first observation is that two of the graphs Borovi\'canin found are the complements of two of the graphs found by Cvetkovi\'c and Simi\'c. Using the Courant-Weyl inequalities we show why this is and reprove the result of Borovi\'canin, highlighting some features of the graphs found by both.
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