# Laplacian Controllability and Observability of Multi-Agent Systems: Recent Advances in Tree Graphs
## 3 Laplacian Controllability and Observability of Starlike Graphs
### Starlike Eigenvalues Associated with Nowhere Zero Eigenvectors  
One interesting and surprising result follows. It basically states that loss of reachability and observability may occur only for those eigenvalues characterizing the grounded Laplacians of the constitutive branch paths. It implies that integer eigenvalues different from one have nowhere zero eigenvectors, and hence they can be controlled by any node.  
**Proposition 1**: _We let \(\Lambda_{S}=\{\lambda\in\mathbb{C}|\det(\lambda I-L_{s})\}\) be the spectrum of a star graph \(S(i_{1},\ldots,i_{s})\), and define \(\Lambda_{P_{S}}=\bigcup_{i=1}^{\theta}\Lambda_{L_{P_{2i+1}}}\), where, for each branch, \(i=1,\ldots,a_{i}\). Any eigenvalue belonging to the set \(\Lambda_{S}-\Lambda_{P_{S}}\) (where \(-\) stands for set difference) is always reachable and observable from any node._  
Proof.: We compute an eigenvector with one zero component. Consider the eigenvalue equation built on a Laplacian matrix as (8) and an eigenvector having zero in any position, namely  
\[\begin{array}{|c|c|c|c|c|c|}\hline a_{11}-\lambda&-1&0&\ldots&\ldots&-1&0& \ldots\\ \hline-1&&&&\\ 0&N_{i_{1}}-\lambda I_{i_{1}}&\ldots&\mathbf{0}_{i_{1}\times i_{\ell}}\\ \vdots&&&&\\ \hline\vdots&\vdots&&&\vdots\\ \hline-1&&&&\\ 0&&&&\mathbf{0}_{i_{\ell}\times i_{1}}&\ldots&N_{i_{\ell}}-\lambda I_{i_{\ell} }\\ \vdots&&&&\\ \hline\end{array}\begin{bmatrix}\mathbf{v}_{1}\\ 0\\ \mathbf{v}_{2}\\ \mathbf{v}_{2}\\ \end{bmatrix}=\mathbf{0}, \tag{13}\]  
where \(\mathbf{v}_{2}\in\mathbb{R}^{a_{\ell}}\) with \(a_{\ell}\leq i_{\ell}\), and both \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) are nonzero. Exploiting (13), one has  
\[\left\{\begin{array}{l}L_{S_{1}}\mathbf{v}_{1}=\lambda\mathbf{v}_{1},\\ (\mathbf{v}_{1})_{1}=(\mathbf{v}_{2})_{1},\\ L_{N_{a_{\ell}}}\mathbf{v}_{2}=\lambda\mathbf{v}_{2},\end{array}\right. \tag{14}\]  
Figure 2: Example of external branch (**above**) and internal branch (**below**). They constitute the basic elements of the proposed graph reduction technique proposed in this paper. Grey nodes are related to nonzero eigenvector components, white double-circled nodes are associated to zero eigenvector components.  
where \(\mathcal{S}_{1}\) is the starlike graph obtained by removing the subpath of length \(a_{\ell}\) from the \(s\)th branch. Last equation \(L_{N_{a_{\ell}}}\mathbf{v}_{2}=\lambda\mathbf{v}_{2}\) implies that, in such a case of eigenvector with one zero component, \(\lambda\) must be necessarily also an eigenvalue of \(N_{a_{\ell}}\), and by property \(P2\) of Proposition A2, \(\lambda\) must be also eigenvalue of \(L_{\mathbb{P}_{2t+1}}\). Since the ordering of the branches is generic, this proof applies to any branch, and the statement follows.  
Coming back to the example depicted in Figure 1, the interested reader can verify that \(\Lambda_{b}\) can never be expressed as an eigenvalue of the constitutive paths, and the whole set of eigenvector is nowhere zero, so that the starlike graph of Figure 1b is reachable and observable from any node of the graph.