exact=true or false

If exact or exact=true is specified, the eigenvalues are returned as exact expressions. If exact=false, the eigenvalues are returned as floating-point expressions. The default is false.

The GraphSpectrum command returns the spectrum of the eigenvalues of the adjacency matrix of a specified graph. The output is a list $L$. Each element of $L$ is a list of size 2, where the first element is an eigenvalue and the second element is its multiplicity.

If argument exact or exact=true is provided, the eigenvalues are returned as exact expressions. Otherwise the eigenvalues are returned as floating-point expressions.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{C5}≔\mathrm{CycleGraph}\left(5\right)$

$\mathrm{C5}≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 5\; edge(s)}$

$\mathrm{GraphSpectrum}\left(\mathrm{C5}\right)$

$\left[\left[\mathrm{-1.618033990}\,2\right]\,\left[0.6180339900\,2\right]\,\left[2.000000000\,1\right]\right]$

$\mathrm{GraphSpectrum}\left(\mathrm{C5}\,\mathrm{exact}\right)$

$\left[\left[-\frac{\sqrt{5}}{2}-\frac{1}{2}\,2\right]\,\left[\frac{\sqrt{5}}{2}-\frac{1}{2}\,2\right]\,\left[2\,1\right]\right]$

$f≔\mathrm{CharacteristicPolynomial}\left(\mathrm{C5}\,x\right)$

$f≔x^5-5x^3+5x-2$

$\mathrm{factor}\left(f\right)$

$\left(x-2\right){\left(x^2+x-1\right)}^2$