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

Lee Gumbrell

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

Combinatorics 2012-04-16 v1

Authors: Lee Gumbrell

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 $\lambda_3\leq0$ . 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.

Keywords

line graph bipartite complement eigenvalue Courant-Weyl inequality graph theory third largest eigenvalue complement

Cite

@article{arxiv.1204.2981,
  title  = {A connection between the bipartite complements of line graphs and the line graphs with two positive eigenvalues},
  author = {Lee Gumbrell},
  journal= {arXiv preprint arXiv:1204.2981},
  year   = {2012}
}

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4 pages, 2 figures

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

Abstract

Keywords

Cite

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