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

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

2012-04-13

arxiv.org/abs/1204.2981v1

Mathematics

Combinatorics

4 pages, 2 figures

Scientific paper

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 $\lambda_3\leq0$. 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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