On an inverse tridiagonal eigenvalue problem and its application to synchronization of network motion

In this work, motivated by the study of stability of the synchronous orbit of a network with tridiagonal Laplacian matrix, we first solve an inverse eigenvalue problem which builds a tridiagonal Laplacian matrix with eigenvalues ${\lambda }_1=0<{\lambda }_2<\cdots <{\lambda }_N$ and null-vector $\mathit{e}=\left[\begin{array}{c}1\\ \mathrm{⋮}\\ 1\end{array}\right]$. Then, we show how this result can be used to guarantee – if possible – that a synchronous orbit of a connected tridiagonal network associated to the matrix $L$ above is asymptotically stable, in the sense of having an associated negative Master Stability Function (MSF). We further show that there are limitations when we also impose symmetry for $L$.