Abstract
If X is a graph with incidence matrix B, then the adjacency matrix of its line graph L(X) is equal to B T B-2I. Because B T B is positive semidefinite, it follows that the minimum eigenvalue of L(X) is at least −2. This chapter is devoted to showing how close this property comes to characterizing line graphs. The main result is a beautiful characterization of all graphs with minimum eigenvalue at least −2. One surprise is that the proof uses several seemingly unrelated combinatorial objects. These include generalized quadrangles with lines of size three and root systems, which arise in connection with a number of important problems, including the classification of Lie algebras.
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© 2001 Springer Science+Business Media New York
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Godsil, C., Royle, G. (2001). Line Graphs and Eigenvalues. In: Algebraic Graph Theory. Graduate Texts in Mathematics, vol 207. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0163-9_12
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DOI: https://doi.org/10.1007/978-1-4613-0163-9_12
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-95220-8
Online ISBN: 978-1-4613-0163-9
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