Spectra of graphs and the spectral criterion for property (T) - Alain Valette (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2017)
For a finite connected graph X, we consider the graph RX obtained from X by associating a new vertex to every edge of X and joining by edges the extremities of each edge of X to the corresponding new vertex. We express the spectrum of the Laplace operator on RX as a function of the corresponding spectrum on X. As a corollary, we show that X is a complete graph if and only if λ_1(RX)>12. We give a re-interpretation of the correspondence XRX in terms of the right-angled Coxeter group defined by X.