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)
Signless Laplacians of finite graphs. (English) Zbl 1113.05061
Summary: We survey properties of spectra of signless Laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix. For regular graphs the whole existing theory of spectra of the adjacency matrix and of the Laplacian matrix transfers directly to the signless Laplacian, and so we consider arbitrary graphs with special emphasis on the non-regular case. The results which we survey (old and new) are of two types: (a) results obtained by applying to the signless Laplacian the same reasoning as for corresponding results concerning the adjacency matrix, (b) results obtained indirectly via line graphs. Among other things, we present eigenvalue bounds for several graph invariants, an interpretation of the coefficients of the characteristic polynomial, a theorem on powers of the signless Laplacian and some remarks on star complements.

05C50Graphs and linear algebra
Full Text: DOI
[1] Chen, Y.: Properties of spectra of graphs and line graphs. Appl. math. J. chinese univ. Ser. B 17, No. 3, 371-376 (2002) · Zbl 1022.05046
[2] Cvetković, D.: Graphs with least eigenvalue - 2: the eigenspace of the eigenvalue - 2. Rendiconti sem. Mat. messina, ser. II 25, No. 9, 63-86 (2003)
[3] Cvetković, D.: Signless Laplacians and line graphs. Bull. acad. Serbe sci. Arts, cl. Sci. math. Natur., sci. Math. 131, No. 30, 85-92 (2005) · Zbl 1119.05066
[4] Cvetković, D.; Doob, M.; Sachs, H.: Spectra of graphs. (1995) · Zbl 0824.05046
[5] Cvetković, D.; Lepović, M.: Cospectral graphs with least eigenvalue at least - 2. Publ. inst. Math. (Beograd) 78, No. 92, 51-63 (2005) · Zbl 1265.05358
[6] Cvetković, D.; Rowlinson, P.; Simić, S.: Eigenspaces of graphs. (1997) · Zbl 0878.05057
[7] Cvetković, D.; Rowlinson, P.; Simić, S.: Spectral generalizations of line graphs, on graphs with least eigenvalue - 2. (2004) · Zbl 1061.05057
[8] Van Dam, E. R.; Haemers, W.: Which graphs are determined by their spectrum?. Linear algebra appl. 373, 241-272 (2003) · Zbl 1026.05079
[9] Dedo, E.: La reconstruibilita del polinomio carrateristico del comutato di un grafo. Boll. unione mat. Ital. 18A, No. 5, 423-429 (1981) · Zbl 0481.05050
[10] Desai, M.; Rao, V.: A characterization of the smallest eigenvalue of a graph. J. graph theory 18, 181-194 (1994) · Zbl 0792.05096
[11] Faria, I.: Permanental roots and the star degree of a graph. Linear algebra appl. 64, 255-265 (1985) · Zbl 0559.05041
[12] Gantmacher, F. R.: Theory of matrices I, II. (1960) · Zbl 0088.25103
[13] Grone, R.; Merris, R.; Sunder, V. S.: The Laplacian spectrum of a graph. SIAM J. Matrix anal. Appl. 11, 218-238 (1990) · Zbl 0733.05060
[14] Gutman, I.; Cvetković, D.: Relations between graphs and special functions. Univ. kragujevac, coll. Sci. papers, fac. Sci. 1, 101-119 (1980)
[15] Haemers, W.; Spence, E.: Enumeration of cospectral graphs. Europ. J. Comb. 25, 199-211 (2004) · Zbl 1033.05070
[16] Lihtenbaum, L. M.: A duality theorem for simple graphs. Usp. mat. Nauk. 18, No. 5, 185-190 (1958)
[17] Lihtenbaum, L. M.: Traces of powers of the vertex- and edge-neighbourhood matrix of a simple graph (Russian). Izv. viss. Ucebn. zav. Mat. 5, 154-163 (1959)
[18] Rowlinson, P.: Star complements in finite graphs: a survey. Rendiconti sem. Mat. messina 8, 145-162 (2002) · Zbl 1042.05061
[19] Simić, S. K.; Marzi, E. M. Li; Belardo, F.: Connected graphs of fixed order and size with maximal index: structural considerations. Le matematiche 59, No. 1 -- 2, 349-365 (2004) · Zbl 1195.05045
[20] E.V. Vahovskii&caron: On the eigenvalues of the neighbourhood matrix of simple graphs (Russian). Sibir. mat. J. 6, 44-49 (1965)