# Laplacian Controllability and Observability of Multi-Agent Systems: Recent Advances in Tree Graphs
## 3 Laplacian Controllability and Observability of Starlike Graphs
### Starlike Trees with Simple Eigenvalues--Preliminary Results  
We start the analysis focusing on starlike graphs having simple eigenvalues only. It is worth mentioning that for trees, integer Laplacian eigenvalues larger than one are necessarily simple [45]. In contrast to this, Laplacian eigenvalue \(\lambda=1\) is often present in the tree Laplacian spectrum, and its occurrence and multiplicity have been longly studied by mathematicians [46].  
In the following sections of the paper, we adopt the labeling described next. The central node, namely the only node having degree larger than two, is usually denoted by subindex \(c\) and it is put as first entry of vector \(\mathbf{x}(t)\). Then, \((\mathbf{x})_{i}(t)\), \(i=2,\ldots,a_{1}+1\) are the values of nodes starting from the neighbor of the central node to the pendant node of Branch 1. The other branches are labeled sequentially and they follow the same ordering logic.  
Considering this labeling, the general form of a starlike graph is  
\[L_{w}\!\!=\!\!\begin{bmatrix}a_{11}&-1&0&\ldots&\ldots&-1&0&\ldots\\ \hline-1&&&&&&&\\ 0&&N_{i_{1}}&&\ldots&\mathbf{0}_{i_{1}\times i_{\ell}}\\ \vdots&&&&&&&\\ \vdots&&\vdots&&&\vdots\\ \hline-1&&&&&&&\\ 0&&\mathbf{0}_{i_{\ell}\times i_{1}}&\ldots&N_{i_{\ell}}\\ \vdots&&&&&&&\\ \end{bmatrix} \tag{8}\]  
where \(a_{11}=s\) is the number of branches starting from the central node. Matrices \(N_{i}\) are some fundamental structured matrices for the forthcoming analysis, and they are defined and characterized as follows.  
We refer to an _external branch_ of \(p\) nodes as the path with one grounded external node, with grounded Laplacian matrix:  
\[N_{p}=\begin{bmatrix}2&-1&&0\\ -1&2&&\\ 0&&\ddots&-1\\ 0&&-1&1\end{bmatrix}, \tag{9}\]  
and to an _internal branch_ of \(q\) nodes as  
\[M_{q}=\begin{bmatrix}2&-1&&0\\ -1&2&&\\ 0&&\ddots&-1\\ 0&&-1&2\end{bmatrix}, \tag{10}\]  
Figure 2 shows an example of an external branch of two nodes (upper side of the Figure 2) and one internal branch of two nodes. The associated grounded Laplacians are  
\[N_{2}=\begin{bmatrix}2&-1\\ -1&1\end{bmatrix}, \tag{11}\]  
and  
\[M_{2}=\begin{bmatrix}2&-1\\ -1&2\end{bmatrix}. \tag{12}\]  
The grounded nodes are graphically denoted with a double circle.  
Laplacian spectrum of path graphs, of the two class of matrices \(N_{i}\), \(M_{j}\), and their relations are reported in Appendix A.2.  
Considering again the example depicted in Figure 1, it is easy to compute from (A7) \(\lambda_{N_{2}}=\frac{3\pm\sqrt{5}}{2}\) and from (A8) \(\lambda_{M_{2}}=3\) and \(\lambda_{M_{2}}=1\). Moreover, it is left to the reader to verify that \(\frac{3\pm\sqrt{5}}{2}\) are also the eigenvalues of \(L_{5}\) using Equation (A5) for \(j=1\) and \(j=3\) and finally that the eigenvalues of \(M_{2}\) are indeed coicident with those of \(L_{3}\) (but the zero eigenvalue).  
A first simple yet important result follows. It is focused on an eigenvector condition that ensures controllability/observability from any node, namely having a nonzero value in each entry. It turns out a special condition on eigenvalues, namely that only eigenvalues associated to subpaths of the branches can be unreachable.