PIGALE 1.3.9 Documentation (original) (raw)

Q distance. Test the distance

$[ d^2(i,j)=\begin{cases} 0,&\text{if <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>=</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i=j</annotation></semantics></math>i=j}\\ 1,&\text{if <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>i and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>j are non adjacent}\\ 1-\frac{1}{\sqrt{d(i)d(j)}},&\text{otherwise} \end{cases}]$

Remark that the matrix ![$ \left(\frac{A_{i,j}}{\sqrt{d(i)d(j)}}\right)_{i,j} ](http://pigale.sourceforge.net/doc/html/form26.png)issymmetric,stochasticandhencehaverealeigenvalues,thelargestonebeingequalto! is symmetric, stochastic and hence have real eigenvalues, the largest one being equal to ![](http://pigale.sourceforge.net/doc/html/form26.png)issymmetric,stochasticandhencehaverealeigenvalues,thelargestonebeingequalto! 1$. Nevertheless, a multigraph is not reconstructible from this distance as replacing every edge by $ k $ parallel edges does not affect the distances between the vertices.