Spectra of graphs and the spectral criterion for property (T)

Alain Valette


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 $\lambda_1(RX)>\frac{1}{2}$. We give a re-interpretation of the correspondence $X\mapsto RX$ in terms of the right-angled Coxeter group defined by $X$.


graph spectrum, Coxeter group, property (T), spectral criterion

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DOI: http://dx.doi.org/10.5614/ejgta.2017.5.1.11


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