Geodesic
The real symmetric matrices of odd order with a P-set of maximum size
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 1007-1026

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

Suppose that $A$ is a real symmetric matrix of order $n$. Denote by $m_A(0)$ the nullity of $A$. For a nonempty subset $\alpha $ of $\{1,2,\ldots ,n\}$, let $A(\alpha )$ be the principal submatrix of $A$ obtained from $A$ by deleting the rows and columns indexed by $\alpha $. When $m_{A(\alpha )}(0)=m_{A}(0)+|\alpha |$, we call $\alpha $ a P-set of $A$. It is known that every P-set of $A$ contains at most $\lfloor {n}/{2} \rfloor $ elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As a first step to the full characterization of these graphs of odd order, we establish some conditions for such graphs $G$ under which there is a real symmetric matrix $A$ whose graph is $G$ and contains a P-set of size ${(n-1)}/{2}$.
DOI : 10.1007/s10587-016-0306-6
Classification : 05C50, 15A18
Keywords: real symmetric matrix; graph; multiplicity of eigenvalues; P-set; P-vertices 
@article{10_1007_s10587_016_0306_6,
     author = {Du, Zhibin and da Fonseca, Carlos M.},
     title = {The real symmetric matrices of odd order with a {P-set} of maximum size},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1007--1026},
     publisher = {mathdoc},
     volume = {66},
     number = {3},
     year = {2016},
     doi = {10.1007/s10587-016-0306-6},
     mrnumber = {3556881},
     zbl = {06644047},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0306-6/}
}
TY  - JOUR
AU  - Du, Zhibin
AU  - da Fonseca, Carlos M.
TI  - The real symmetric matrices of odd order with a P-set of maximum size
JO  - Czechoslovak Mathematical Journal
PY  - 2016
SP  - 1007
EP  - 1026
VL  - 66
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0306-6/
DO  - 10.1007/s10587-016-0306-6
LA  - en
ID  - 10_1007_s10587_016_0306_6
ER  - 
%0 Journal Article
%A Du, Zhibin
%A da Fonseca, Carlos M.
%T The real symmetric matrices of odd order with a P-set of maximum size
%J Czechoslovak Mathematical Journal
%D 2016
%P 1007-1026
%V 66
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0306-6/
%R 10.1007/s10587-016-0306-6
%G en
%F 10_1007_s10587_016_0306_6
Du, Zhibin; da Fonseca, Carlos M. The real symmetric matrices of odd order with a P-set of maximum size. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 1007-1026. doi: 10.1007/s10587-016-0306-6

Cité par Sources :