Eigenvalue bounds for the signless Laplacian. (English) Zbl 1164.05038

Cvetković, Dragoš; Rowlinson, Peter; Simić, Slobodan K.

Publ. Inst. Math., Nouv. Sér. 81(95), 11-27 (2007).

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.
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.

Reviewer: Zoran Stanić (Beograd)

MSC:

05C50

Graphs and linear algebra (matrices, eigenvalues, etc.)

Keywords:

signless Laplacian eigenvalues; line graphs; largest eigenvalue