zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

05C50Graphs and linear algebra