ELibM Journals • ELibM Home • EMIS Home • EMIS Mirrors

  EMIS Electronic Library of Mathematics (ELibM)
The Open Access Repository of Mathematics
  EMIS ELibM Electronic Journals

JOURNAL OF
ALGEBRAIC
COMBINATORICS

  Editors-in-chief: C. A. Athanasiadis, T. Lam, A. Munemasa, H. Van Maldeghem
ISSN 0925-9899 (print) • ISSN 1572-9192 (electronic)
 

Adjacency preservers, symmetric matrices, and cores

Marko Orel

DOI: 10.1007/s10801-011-0318-0

Abstract

It is shown that the graph Γ  n that has the set of all n\times  n symmetric matrices over a finite field as the vertex set, with two matrices being adjacent if and only if the rank of their difference equals one, is a core if n\geq 3. Eigenvalues of the graph Γ  n are calculated as well.

Pages: 633–647

Keywords: keywords core; adjacency preserver; symmetric matrix; finite field; eigenvalue of a graph; coloring; quadratic form

Full Text: PDF

References

1. Biggs, N.: Algebraic Graph Theory. In: Cambridge Tracts in Mathematics, No.
67. Cambridge University Press, London (1974)
2. Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Electronic book available at . Accessed 12 September 2010
3. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Ergeb. Math. Ihrer Grenzgeb. 3, 18 (1989)
4. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Additions and Corrections to the book `Distance-Regular Graphs'. Available at Accessed 9 April 2011
5. Cameron, P.J., Kazanidis, P.A.: Cores of symmetric graphs. J. Aust. Math. Soc. 85(2), 145-154 (2008)
6. Delsarte, P.: Bilinear forms over a finite field with applications to coding theory. J. Comb. Theory, Ser. A 25(3), 226-241 (1978)
7. Delsarte, P., Goethals, J.M.: Alternating bilinear forms over GF (q). J. Comb. Theory, Ser. A 19(1), 26-50 (1975)
8. Feng, R., Wang, Y., Ma, C., Ma, J.: Eigenvalues of association schemes of quadratic forms. Discrete Math. 308(14), 3023-3047 (2008)
9. Gao, J.-F., Wan, Z.-X., Feng, R.-Q., Wang, D.-J.: Geometry of symmetric matrices and its applications III. Algebra Colloq. 3(2), 135-146 (1996)
10. Godsil, C.D.: Algebraic Combinatorics. Chapman and Hall Mathematics Series. Chapman & Hall, New York (1993)
11. Godsil, C.D., Royle, G.F.: Algebraic Graph Theory. In: Graduate Texts in Mathematics, vol.
207. Springer, New York (2001)
12. Godsil, C.D., Royle, G.F.: Cores of geometric graphs. (2008)
13. Guterman, A., Li, C.-K., Šemrl, P.: Some general techniques on linear preserver problems. Linear Algebra Appl. 315(1-3), 61-81 (2000)
14. Hell, P., Nešet\check ril, J.: Graphs and Homomorphisms. Oxford Lecture Series in Mathematics and its Applications, vol.
28. Oxford University Press, Oxford (2004)
15. Hoffman, A.J.: On eigenvalues and colorings of graphs. Graph theory and its applications. In: Proc. Advanced Sem., Math. Research Center, Univ. of Wisconsin, Madison, Wis, 1969, pp. 79-91. Academic Press, New York (1970)
16. Hua, L.-K.: Geometry of symmetric matrices over any field with characteristic other than two. Ann. Math. 50(1), 8-31 (1949)
17. Huang, L.-P.: Adjacency preserving bijection maps of Hermitian matrices over any division ring with an involution. Acta Math. Sin. Engl. Ser. 23(1), 95-102 (2007)
18. Huang, L.-P.: Geometry of n \times n(n \geq 3) Hermitian matrices over any division ring with an involution and its applications. Commun. Algebra 36(6), 2410-2438 (2008)
19. Huang, W.-L.: Adjacency preserving mappings of 2 \times 2 Hermitian matrices. Aequ. Math. 75(1-2), 51-64 (2008)
20. Huang, W.-L., Šemrl, P.: Adjacency preserving maps on hermitian matrices. Can. J. Math. 60(5), 1050-1066 (2008)
21. Huang, W.-L., Höfer, R., Wan, Z.-X.: Adjacency preserving mappings of symmetric and hermitian matrices. Aequ. Math. 67(1-2), 132-139 (2004)
22. Imrich, W., Klavžar, S.: Product Graphs, Structure and Recognition. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (2000)
23. Legiša, P.: Adjacency preserving mappings on real symmetric matrices. Math. Commun. (2011, in press).
24. Li, C.-K., Pierce, S.: Linear preserver problems. Am. Math. Mon. 108(7), 591-605 (2001)
25. Lidl, R., Niederreiter, H.: Finite Fields. With a foreword by P. M. Cohn. In: Encyclopedia of Mathematics and its Applications, vol.
20. Cambridge University Press, Cambridge (1987)
26. Orel, M.: A note on adjacency preservers on hermitian matrices over finite fields. Finite Fields Appl. 15(4), 441-449 (2009)
27. Šemrl, P.: Maps on matrix spaces. Linear Algebra Appl. 413(2-3), 364-393 (2006)
28. Šemrl, P.: Linear Preserver Problems. In: Hogben, L. (ed.), Handbook of Linear Algebra. Discrete




© 1992–2009 Journal of Algebraic Combinatorics
© 2012 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition