A d'Alembert type formula and spectrum of the Laplacian on a graph with commensurable edges (Q2780408)

This paper is devoted to the Cauchy problem on a graph \(\Gamma\), that is NEWLINE\[NEWLINE\begin{cases}{\partial^2 u\over\partial t^2}=\Delta u,\;t\geq 0,\\ u(0,x)=f(x),\\ {\partial u\over\partial t}(0,x)=0.\end{cases}\tag{1}NEWLINE\]NEWLINE The authors find a d'Alembert type formula for the solution of (1) and based on this formula construct a matrix \(W\) connecting the values of solution of (1) for \(t=1\) with the initial values \(f(x)\). Under suitable assumptions on \(f\) and the graph \(\Gamma\) the authors study the spectrum of \(-\Delta\) on \(\Gamma\).