Colin de Verdière graph invariant

Colin de Verdière's invariant is a graph parameter ${\displaystyle \mu (G)}$ for any graph G, introduced by Yves Colin de Verdière in 1990. It was motivated by the study of the maximum multiplicity of the second eigenvalue of certain Schrödinger operators.

Let ${\displaystyle G=(V,E)}$ be a loopless simple graph. Assume without loss of generality that ${\displaystyle V=\{1,\dots ,n\}}$. Then ${\displaystyle \mu (G)}$ is the largest corank of any symmetric matrix ${\displaystyle M=(M_{i,j})\in \mathbb {R} ^{(n)}}$ such that:

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