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)
Some results on graph spectra. (English) Zbl 1015.05051
Summary: This paper presents a variety of results on graph spectra. The number of main eigenvalues of a graph is shown to be equal to the rank of an associated matrix. We establish a condition for a graph to have exactly two main eigenvalues and then show how to evaluate them and their associated eigenvectors. It is shown that the main eigenvalues and corresponding eigenvectors of a graph determine those of its complement. We generalize to any eigenvalue a condition for $0$ and $-1$ to be eigenvalues of a graph and its complement, respectively. Finally, we generalize to non-simple eigenvalues a result on the components of an eigenvector associated with a simple eigenvalue.

MSC:
05C50Graphs and linear algebra
WorldCat.org
Full Text: DOI
References:
[1] Bell, F. K.; Rowlinson, P.: Certain graphs without zero as an eigenvalue. Math. japonica 48, 961-967 (1993) · Zbl 0792.05094
[2] Cvetković, D. M.: Graphs and their spectra. Univ. beograd, publ. Elektrotehn fak., ser. Mat. fiz. 354--356, 1-50 (1971)
[3] Cvetković, D. M.: The Main part of the spectrum, divisors and switching of graphs. Publ. inst. Math. (Beograd) 23, No. 37, 31-38 (1978) · Zbl 0423.05028
[4] Cvetković, D.; Doob, M.: Developments in the theory of graph spectra. Linear and multilinear algebra 18, 153-181 (1985) · Zbl 0615.05039
[5] Cvetković, D.; Fowler, P. W.: A group-theoretical bound for the number of Main eigenvalues of a graph. J. chem. Inf. comput. Sci. 39, 638-641 (1999)
[6] Cvetković, D.; Rowlinson, P.; Simić, S.: Eigenspaces of graphs. Encyclopedia of mathematics and its applications 66 (1997) · Zbl 0878.05057
[7] Harary, F.; Schwenk, A. J.: The spectral approach to determining the number of walks in a graph. Pacific J. Math. 80, 443-449 (1979) · Zbl 0417.05032
[8] Li, Q.; Feng, K. Q.: On the largest eigenvalue of a graph. Acta math. Appl. sinica 2, 167-175 (1979)
[9] Mukherjee, A. K.; Datta, K. K.: Two new graph-theoretical methods for the generation of eigenvectors of chemical graphs. Proc. indian acad. Sci. chem. Sci. 101, 499-517 (1989)
[10] Powers, D. L.; Sulaiman, M. M.: The walk partition and colorations of graphs. Linear algebra appl. 48, 145-159 (1982) · Zbl 0501.05044