# Laplacian Controllability and Observability of Multi-Agent Systems: Recent Advances in Tree Graphs
## 4 Zero-Nonzero Pattern of Eigenvectors Associated to \(\lambda=1\)  
As per the previous result, we start the investigation by considering subpaths of incresing dimension. In this Section, we thoroughly discuss the case \(\lambda=1\). The corresponding matrix \(N_{1}\) is the scalar value \(N_{1}=1\), so that its only eigenvalue is \(\lambda=1\). It is worth noting that the eigenstructure of \(\lambda=1\) for tree graphs is widely considered in the literature from the seminal work of [45] to our days [47], and it is still a debated topic. In the following, we provide a description of the two different conditions that make \(\lambda=1\) a Laplacian eigenvalue of a star-like graph \(\mathcal{S}(a_{1},\ldots,a_{s})\). The first stage is the reduction in the star-like graph to its _core_, namely a subgraph obtained by reducing the branches to a 'core' of few nodes by reducing from the pendant node each branch as described analitically in Appendix B, or graphically, as depicted in Figure 3. Figure 4 shows an example of reduction to a core. Figure 4: Example of a graph reduction to a core related to \(N_{1}\), related to \(\lambda=1\). The core graph is the subgraph made of \(v_{\ell},v_{1},v_{2},v_{3},v_{4},v_{5}\). Figure 3: Zero pattern associated to \(N_{1}\). The upper and middle branches constitute the basic elements of the proposed graph reduction technique in case of \(\lambda=1\) (external and internal, respectively); the lowest is an example of zero-nonzero pattern of eigenvector starting from one leaf, put on the left. Grey nodes are related to nonzero eigenvector components, double-circled white nodes are associated to zero eigenvector components. We now focus on a generic core to find conditions under which the considered Laplacian admits \(\lambda=1\) as eigenvalue, or eventually not. A thorough investigation by simulation on all possible core structures provided two different topology conditions coherent with \(\lambda=1\) at the central node, which are briefly described in the following. We denote by \((\mathbf{v})_{1 pre^{\prime}}\), \((\mathbf{v})_{2 pre^{\prime}}\)... the values afforded by the neighbors of \((\mathbf{v})_{1}\), \((\mathbf{v})_{2}\),... at the opposite position with respect to \((\mathbf{v})_{c}\) applying the reduction technique described in Appendix B. The two conditions consistent with \(\lambda=1\) at the central node are as follows:  
1. If any of the branch reductions show \((\mathbf{v})_{i}=(\mathbf{v})_{ipre}\), then at least one other should show analogously \((\mathbf{v})_{j}=(\mathbf{v})_{jpre^{\prime}}\), otherwise \(\lambda=1\) is not an eigenvalue of \(\mathcal{S}\). This is the case when \((\mathbf{v})_{c}=0\). In this case, we denote by \(r\) the number of branches whose reduction satisfies \((\mathbf{v})_{c}=0\), \(m_{g}(1)=r-1\). 2. If all \((\mathbf{v})_{i}\
eq(\mathbf{v})_{ipre^{\prime}}\) then the eigenvalue condition (see, e.g., Equation (A9)) evaluated at the central node has solution if and only if only one \((\mathbf{v})_{i}=0\) and all the other \((\mathbf{v})_{j}=a\
eq 0\), for any \(j\
eq i\). In this case, the solution is \((\mathbf{v})_{c}=a\), and whenever the above condition holds, in this case \(m_{g}(1)=1\) always. To provide an example, we come back to the graph depicted in Figure 4. It is possible to see that node evaluation matches the condition \((\mathbf{v})_{i}\
eq(\mathbf{v})_{ipre^{\prime}}\) so we fall in (R2). However, in this case, two nodes have \((\mathbf{v})_{i}=0\) (namely \((\mathbf{v})_{1}=0\) and \((\mathbf{v})_{3}=0\) ), so that (R2) is not satisfied, and this means that \(\lambda=1\) is not an eigenvalue of \(\mathcal{S}(2,3,2,3,6)\). Indeed, in this case, it is possible to verify that the eigenvalue equation computed at the central node does not hold. Figure 5 shows some graphs with few differences from Figure 4, namely \(\mathcal{S}(2,3,3,3,6)\) and \(\mathcal{S}(2,4,3,4,7)\), but they both have \(\lambda=1\) in their spectrum and different multiplicity. Consider Figure 5a; in this case, Condition (R2) is fully met.