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

Czechoslovak Mathematical Journal

Carlos M. da Fonseca¹  Zhibin Du² 

[1] Department of Mathematics, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait City, Kuwait;School of Mathematics and Statistics, Zhaoqing University, Xinghu Ave, Zhaoqing 526061, Duanzhou, Guangdong, China

关键词: real symmetric matrix;    graph;    multiplicity of eigenvalues;    P-set;    P-vertices;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

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 ${(n-1)}/2$.

【 授权许可】

Unknown