We first give a result on eigenvalues of the line graph of a graph. We then use the result to present a new upper bound for eigenvalues of the Laplacian matrix of a graph. Moreover we determine all graphs the largest eigenvalue of whose Laplacian matrix reaches the upper bound.
A lower bound for the Laplacian eigenvalues of a graph—Proof of a conjecture by Guo
✍ Scribed by Andries E. Brouwer; Willem H. Haemers
- Publisher
- Elsevier Science
- Year
- 2008
- Tongue
- English
- Weight
- 97 KB
- Volume
- 429
- Category
- Article
- ISSN
- 0024-3795
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We prove that the minimum value of the least eigenvalue of the signless Laplacian of a connected nonbipartite graph with a prescribed number of vertices is attained solely in the unicyclic graph obtained from a triangle by attaching a path at one of its endvertices.
In the note, we present an upper bound for the spectral radius of Laplacian matrix of a graph in terms of a "2-degree" of a vertex.
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