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

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.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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