On the core-nilpotent decomposition of trees

JAUME, DANIEL ALEJANDRO; SOTA, RODRIGO

Niterói

Workshop; LAAW 2018: Linear Algebra and its Applications Workshop; 2018

ILAS y Instituto de Matemática e Estatística ? IME, Universidade Federal Fluminense

In this work we show, through the null decomposition of trees given by Jaume and Molina (2018), that the core-nilpotent decomposition of \(A(T)\), the adjacency matrix of a tree \(T\), can be obtained directly from the tree itself. In other words, we give two invertible matrices \(Q\) and \(C\), expressed in terms of the adjacency relations of \(T\), such that \(Q^{-1}A(T)Q\) is a \(2\times 2\) blocks diagonal matrix, whose first block is \(C\) (a \(r \times r\) matrix such that \(\rank{C}=\rank{A(T)}=r\)) and whose second block is a zero matrix. Using the results given by Jaume \textit{et al.} (2018), the core-nilpotent decomposition of trees can be obtained in linear time.