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description Journal article public Electronic Journal of Graph Theory and Applications

Spectra of Graphs and the Spectral Criterion for Property (T)

Alain Valette
Diterbitkan 2017

Abstrak

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 $X$.

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