On eigenvectors of mixed graphs with exactly one nonsingular cycle

Yi-Zheng Fan

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 4, page 1215-1222
  • ISSN: 0011-4642

Abstract

top
Let G be a mixed graph. The eigenvalues and eigenvectors of G are respectively defined to be those of its Laplacian matrix. If G is a simple graph, [M. Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory, Czechoslovak Math. J. 25 (1975), 619–633] gave a remarkable result on the structure of the eigenvectors of G corresponding to its second smallest eigenvalue (also called the algebraic connectivity of G ). For G being a general mixed graph with exactly one nonsingular cycle, using Fiedler’s result, we obtain a similar result on the structure of the eigenvectors of G corresponding to its smallest eigenvalue.

How to cite

top

Fan, Yi-Zheng. "On eigenvectors of mixed graphs with exactly one nonsingular cycle." Czechoslovak Mathematical Journal 57.4 (2007): 1215-1222. <http://eudml.org/doc/31189>.

@article{Fan2007,
abstract = {Let $G$ be a mixed graph. The eigenvalues and eigenvectors of $G$ are respectively defined to be those of its Laplacian matrix. If $G$ is a simple graph, [M. Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory, Czechoslovak Math. J. 25 (1975), 619–633] gave a remarkable result on the structure of the eigenvectors of $G$ corresponding to its second smallest eigenvalue (also called the algebraic connectivity of $G$). For $G$ being a general mixed graph with exactly one nonsingular cycle, using Fiedler’s result, we obtain a similar result on the structure of the eigenvectors of $G$ corresponding to its smallest eigenvalue.},
author = {Fan, Yi-Zheng},
journal = {Czechoslovak Mathematical Journal},
keywords = {mixed graphs; Laplacian eigenvectors; mixed graphs; Laplacian eigenvectors},
language = {eng},
number = {4},
pages = {1215-1222},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On eigenvectors of mixed graphs with exactly one nonsingular cycle},
url = {http://eudml.org/doc/31189},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Fan, Yi-Zheng
TI - On eigenvectors of mixed graphs with exactly one nonsingular cycle
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 4
SP - 1215
EP - 1222
AB - Let $G$ be a mixed graph. The eigenvalues and eigenvectors of $G$ are respectively defined to be those of its Laplacian matrix. If $G$ is a simple graph, [M. Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory, Czechoslovak Math. J. 25 (1975), 619–633] gave a remarkable result on the structure of the eigenvectors of $G$ corresponding to its second smallest eigenvalue (also called the algebraic connectivity of $G$). For $G$ being a general mixed graph with exactly one nonsingular cycle, using Fiedler’s result, we obtain a similar result on the structure of the eigenvectors of $G$ corresponding to its smallest eigenvalue.
LA - eng
KW - mixed graphs; Laplacian eigenvectors; mixed graphs; Laplacian eigenvectors
UR - http://eudml.org/doc/31189
ER -

References

top
  1. 10.1080/03081089808818590, Linear Multilinear Algebra 45 (1998), 247–273. (1998) MR1671627DOI10.1080/03081089808818590
  2. 10.1080/03081089908818623, Linear Multilinear Algebra 46 (1999), 299–312. (1999) MR1729196DOI10.1080/03081089908818623
  3. 10.1080/03081080008818646, Linear Multilinear Algebra 47 (2000), 217–229. (2000) MR1785029DOI10.1080/03081080008818646
  4. Graph Theory with Applications, Elsevier, New York, 1976. (1976) MR0411988
  5. Spectral Graph Theory. Conference Board of the Mathematical Science, No.  92, Am. Math. Soc., Providence, 1997. (1997) MR1421568
  6. 10.1016/S0024-3795(03)00575-5, Linear Algebra Appl. 374 (2003), 307–316. (2003) MR2008794DOI10.1016/S0024-3795(03)00575-5
  7. Algebraic connectivity of graphs, Czechoslovak Math.  J. 23 (1973), 298–305. (1973) Zbl0265.05119MR0318007
  8. A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory, Czechoslovak Math.  J. 25 (1975), 619–633. (1975) MR0387321
  9. Algebraic graph theory without orientations, Linear Algebra Appl. 212/213 (1994), 289–308. (1994) MR1306983
  10. A relation between the matching number and the Laplacian spectrum of a graph, Linear Algebra Appl. 325 (2001), 71–74. (2001) MR1810095
  11. Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985. (1985) MR0832183
  12. 10.1080/03081089808818554, Linear Multilinear Algebra 44 (1998), 131–148. (1998) MR1674228DOI10.1080/03081089808818554
  13. 10.1080/03081089608818448, Linear Multilinear Algebra 40 (1996), 311–325. (1996) MR1384650DOI10.1080/03081089608818448
  14. 10.1002/(SICI)1097-0118(199908)31:4<267::AID-JGT1>3.3.CO;2-4, J.  Graph Theory 31 (1999), 267–273. (1999) MR1698744DOI10.1002/(SICI)1097-0118(199908)31:4<267::AID-JGT1>3.3.CO;2-4
  15. Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197/198 (1994), 143–176. (1994) Zbl0802.05053MR1275613
  16. Some applications of Laplacian eigenvalues of graphs, In: Graph Symmetry, G. Hahn and G. Sabidussi (eds.), Kluwer, Dordrecht, 1997, pp. 225–275. (1997) MR1468791
  17. The Laplacian spectrum of a mixed graph, Linear Algebra Appl. 353 (2002), 11–20. (2002) Zbl1003.05073MR1918746

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.