##### Sesión Matemática Discreta

Diciembre 14, 11:00 ~ 11:20

## Link operation and nullity of trees.

### Jaume, Daniel A.

t{ tttGiven two disjoint trees $T_{1}$ and $T_{2}$, and two vertices, $v_{1} \in V(T_{1})$ and $v_{2} \in V(T_{2})$, the \textbf{link between} $T_{1}$ and $T_{2}$ \textbf{through} $v_{1}$ and $v_{2}$, denoted by ttt$ttt(T_{1},v_{1}) \multimap(T_{2},v_{2}), ttt$ tttis the tree obtained by adding an edge between $v_{1}$ and $v_{2}$: ttt\begin{itemize} tttt\item$V\left((T_{1},v_{1}) \multimap (T_{2},v_{2})\right) = V(T_{1}) \cup V(T_{2})$. tttt\item $E\left((T_{1},v_{1}) \multimap (T_{2},v_{2}) \right) = E(T_{1}) \cup E(T_{2}) \cup \{v_{1},v_{2}\}$. ttt\end{itemize} tt ttWith $T_{1} \multimap T_{2}$ we denote an arbitrary link between both trees. t} The nullity of the adjacency matrix of $T$ is denoted by $null(T)$. The vertices in the support of $T$ are called supported-vertices, see \cite{Jaume2015NDT}. Teorema 1: Let $T_{1}$ and $T_{2}$ be two disjoint trees. If the linked vertices are both supported, then t$tnull(T_{1}\multimap \, T_{2}) = null(T_{1}) + null(T_{2})-2, t$ totherwise t$tnull(T_{1} \multimap \, T_{2}) = null(T_{1}) + null(T_{2}). t$ As direct corollaries we have Corollary 1: Let $T_{1}$ and $T_{2}$ be two disjoint trees. If the linked vertices are both supported, then t$\nu (T_{1} \multimap \, T_{2}) = \nu(T_{1}) +\nu(T_{2})+1,$ totherwise t%\begin{enumerate} t$\nu (T_{1} \multimap \, T_{2}) = \nu(T_{1} +\nu(T_{2}),$ twhere $\nu (T)$ is the matching number of $T$. Corollary 2: Let $T_{1}$ and $T_{2}$ be two disjoint trees. If the linked vertices are both supported, then t%\begin{enumerate} t%t\item $\nu (T_{1} \link \, T_{2}) = \nu(T_{1} +\nu(T_{2})+1$. t$\alpha (T_{1} \multimap \, T_{2})= \alpha (T_{1}) + \alpha (T_{2})-1,$ t%\end{enumerate} totherwise t%\begin{enumerate} t%t\item $\nu (T_{1} \link \, T_{2}) = \nu(T_{1} +\nu(T_{2})$. t$\alpha (T_{1} \multimap \, T_{2}) = \alpha (T_{1}) + \alpha (T_{2}),$ t%\end{enumerate} twhere $\alpha (T)$ is the independent number of $T$. \begin{thebibliography}{9} t \bibitem{Jaume2015NDT}{ Daniel A. Jaume, and Gonzalo Molina. \textit{Null decomposition of trees}. Preprint, 2017.} \end{thebibliography}

Autores: Jaume, Daniel A..