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Zbl 1164.05038
Cvetković, Dragoš; Rowlinson, Peter; Simić, Slobodan K.
Eigenvalue bounds for the signless Laplacian. (English)
[J] Publ. Inst. Math., Nouv. Sér. 81(95), 11-27 (2007). ISSN 0350-1302

The signless Laplacian spectrum of a graph is a spectrum of the matrix $Q=A+D$, where $A$ is its adjacency matrix, while $D$ is the diagonal matrix of vertex degrees. Usually the signless Laplacian matrix is called $Q$-matrix, while its spectrum and eigenvalues are known as the $Q$-spectrum and the $Q$-eigenvalues, respectively. At the begin, the authors extend their previous survey of properties of $Q$-spectrum. The paper also contains a number of computer-generated conjectures. Mostly, the conjectures give some bounds for the first, the second or the least $Q$-eigenvalue of an arbitrary graph. Some comments on the conjectures are given.\par In further, the authors prove their main result: among the connected graphs with fixed order and size, the graph with maximal $Q$-index is a nested split graph. Using this result two conjectures are confirmed. Finally some other conjectures are resolved.
[Zoran Stanić (Beograd)]

MSC 2000:: *05C50 Graphs and matrices

Keywords: signless Laplacian eigenvalues; line graphs; largest eigenvalue

Cited in: Zbl 1237.05124 Zbl 1223.05171 Zbl pre05649343

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