Main eigenvalues of real symmetric matrices with application to signed graphs

Zoran Stanić

Czechoslovak Mathematical Journal (2020)

  • Volume: 70, Issue: 4, page 1091-1102
  • ISSN: 0011-4642

Abstract

top
An eigenvalue of a real symmetric matrix is called main if there is an associated eigenvector not orthogonal to the all-1 vector . Main eigenvalues are frequently considered in the framework of simple undirected graphs. In this study we generalize some results and then apply them to signed graphs.

How to cite

top

Stanić, Zoran. "Main eigenvalues of real symmetric matrices with application to signed graphs." Czechoslovak Mathematical Journal 70.4 (2020): 1091-1102. <http://eudml.org/doc/296986>.

@article{Stanić2020,
abstract = {An eigenvalue of a real symmetric matrix is called main if there is an associated eigenvector not orthogonal to the all-1 vector $\{\bf j\}$. Main eigenvalues are frequently considered in the framework of simple undirected graphs. In this study we generalize some results and then apply them to signed graphs.},
author = {Stanić, Zoran},
journal = {Czechoslovak Mathematical Journal},
keywords = {main angle; signed graph; adjacency matrix; Laplacian matrix; Gram matrix},
language = {eng},
number = {4},
pages = {1091-1102},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Main eigenvalues of real symmetric matrices with application to signed graphs},
url = {http://eudml.org/doc/296986},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Stanić, Zoran
TI - Main eigenvalues of real symmetric matrices with application to signed graphs
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 4
SP - 1091
EP - 1102
AB - An eigenvalue of a real symmetric matrix is called main if there is an associated eigenvector not orthogonal to the all-1 vector ${\bf j}$. Main eigenvalues are frequently considered in the framework of simple undirected graphs. In this study we generalize some results and then apply them to signed graphs.
LA - eng
KW - main angle; signed graph; adjacency matrix; Laplacian matrix; Gram matrix
UR - http://eudml.org/doc/296986
ER -

References

top
  1. Cardoso, D. M., Sciriha, I., Zerafa, C., 10.1016/j.laa.2009.07.039, Linear Algebra Appl. 432 (2010), 2399-2408. (2010) Zbl1217.05136MR2599869DOI10.1016/j.laa.2009.07.039
  2. Cvetković, D., Doob, M., Sachs, H., Spectra of Graphs: Theory and Applications, J. A. Barth Verlag, Heidelberg (1995). (1995) Zbl0824.05046MR1324340
  3. Cvetković, D., Rowlinson, P., Simić, S., 10.1017/CBO9780511801518, London Mathematical Society Student Texts 75, Cambridge University Press, Cambridge (2010). (2010) Zbl1211.05002MR2571608DOI10.1017/CBO9780511801518
  4. Deng, H., Huang, H., 10.13001/1081-3810.1659, Electron. J. Linear Algebra 26 (2013), 381-393. (2013) Zbl1282.05109MR3084649DOI10.13001/1081-3810.1659
  5. Doob, M., A geometric interpretation of the least eigenvalue of a line graph, Combinatorial Mathematics and its Applications R. C. Bose, T. A. Dowling University of North Carolina, Chapel Hill (1970), 126-135. (1970) Zbl0209.55403MR0268060
  6. Haynsworth, E. V., 10.6028/jres.063B.009, J. Res. Natl. Bur. Stand., Sec. B 63 (1959), 73-78. (1959) Zbl0090.24104MR0109432DOI10.6028/jres.063B.009
  7. Hou, Y., Tang, Z., Shiu, W. C., 10.1016/j.aml.2011.11.025, Appl. Math. Lett. 25 (2012), 1274-1278. (2012) Zbl1248.05112MR2947393DOI10.1016/j.aml.2011.11.025
  8. Hou, Y., Zhou, H., Trees with exactly two main eigenvalues, J. Nat. Sci. Hunan Norm. Univ. 28 (2005), 1-3 Chinese. (2005) Zbl1109.05071MR2240441
  9. Petersdorf, M., Sachs, H., Über Spektrum, Automorphismengruppe und Teiler eines Graphen, Wiss. Z. Tech. Hochsch. Ilmenau 15 (1969), 123-128 German. (1969) Zbl0199.27504MR0269552
  10. Rowlinson, P., 10.2298/AADM0702445R, Appl. Anal. Discrete Math. 1 (2007), 445-471. (2007) Zbl1199.05241MR2355287DOI10.2298/AADM0702445R
  11. Stanić, Z., 10.1017/CBO9781316341308, London Mathematical Society Lecture Note Series 423, Cambridge University Press, Cambridge (2015). (2015) Zbl1368.05001MR3469535DOI10.1017/CBO9781316341308
  12. Stanić, Z., 10.1016/j.laa.2019.03.011, Linear Algebra Appl. 573 (2019), 80-89. (2019) Zbl1411.05109MR3933292DOI10.1016/j.laa.2019.03.011
  13. Zaslavsky, T., Matrices in the theory of signed simple graphs, Advances in Discrete Mathematics and Applications B. D. Acharya, G. O. H. Katona, J. Nešetřil Ramanujan Mathematical Society Lecture Notes Series 13, Ramanujan Mathematical Society, Mysore (2010), 207-229. (2010) Zbl1231.05120MR2766941

NotesEmbed ?

top

You must be logged in to post comments.