The real symmetric matrices of odd order with a P-set of maximum size

Du, Zhibin ; da Fonseca, Carlos M.

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 1007-1026.

Résumé

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 $\{1,2,\dots ,n\}$, 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 $(n-1)/2$.

Détail

DOI : 10.1007/s10587-016-0306-6

Classification : 05C50, 15A18
Mots-clés : real symmetric matrix; graph; multiplicity of eigenvalues; P-set; P-vertices