# zbMATH — the first resource for mathematics

##### Examples
 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.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 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)
Exact bounds on the order of the maximum clique of a graph. (English) Zbl 1051.68112

Summary: The paper reviews some of the existing exact bounds to the maximum clique of a graph and successively presents a new upper and a new lower bound. The new upper bound is $\omega ⩽n$ – rank $\overline{A}/2$ , where $\overline{A}$ is the adjacency matrix of the complementary graph, and derives from a formulation of the maximum clique problem in complex space. The new lower bound is $\omega ⩾1/\left(1-{g}_{{j}^{*}}\left({\alpha }^{*}\right)\right)$ (see text for details) and improves strictly the present best lower bound published by H. S. Wilf [J. Comb. Theory, Ser. B 40, 113–117 (1986; Zbl 0598.05047)].

Throughout the paper an eye is kept on the computational complexity of actually calculating the bounds. At the end, the various bounds are compared on 700 random graphs.

##### MSC:
 68R10 Graph theory in connection with computer science (including graph drawing) 05C50 Graphs and linear algebra 05C69 Dominating sets, independent sets, cliques
##### Keywords:
Graph; Maximum clique; Bound