Conformally Rigid Graphs

Abstract

Given a finite, simple, connected graph $G=\left(V,E\right)$ with $\mid V\mid =n$, we consider the associated graph Laplacian matrix $L=D-A$ with eigenvalues $0={\lambda }_1<{\lambda }_2\le \cdots \le {\lambda }_n$. One can also consider the same graph equipped with positive edge weights $w:E\to R_{>0}$ normalized to ${\sum }_{e\in E}{w}_e=\mid E\mid$ and the associated weighted Laplacian matrix $L_w$. We say that $G$ is conformally rigid if constant edge-weights maximize the second eigenvalue ${\lambda }_2\left(w\right)$ of $L_w$ over all $w$, and minimize ${\lambda }_n\left(w^{\prime }\right)$ of $L_{w^{\prime }}$ over all $w^{\prime }$, that is, for all $w,w^{\prime }$,

${\lambda }_2\left(w\right)\le {\lambda }_2\left(1\right)\le {\lambda }_n\left(1\right)\le {\lambda }_n\left(w^{\prime }\right).$

Conformal rigidity requires an extraordinary amount of structure in $G$. Every edge-transitive graph is conformally rigid. We prove that every distance regular graph, and hence every strongly regular graph, is conformally rigid. Certain special graph embeddings can be used to characterize conformal rigidity. Cayley graphs can be conformally rigid but need not be, we prove a sufficient criterion. We also find a small set of conformally rigid graphs that do not belong to any of the above categories; these include the Hoffman graph, the crossing number graph 6B and others. Conformal rigidity can be certified via semidefinite programming, we provide explicit examples.