Conformally rigid graphs

TL;DR

We introduce conformal rigidity for graphs, defining when edge-weight perturbations with fixed total weight cannot improve the spectral extremals $\lambda_2$ and $\lambda_n$ beyond the unweighted case. The main results establish that all edge-transitive and all distance-regular graphs are conformally rigid, with broader evidence that many Cayley and circulant graphs also satisfy the property under a computable criterion; several sporadic outliers are identified. A practical SDP framework with symmetry reduction and dual certificates is developed to certify conformal rigidity and to realize edge-isometric spectral embeddings on the critical eigenspaces, enabling rigorous certification from numerical data. The work connects spectral optimization, graph embeddings, and semidefinite programming to reveal a rich landscape of highly symmetric graphs whose spectral extremality is already realized in the unweighted form, with potential implications for graph sparsification and network design.

Abstract

Given a finite, simple, connected graph $G=(V,E)$ with $|V|=n$, we consider the associated graph Laplacian matrix $L = D - A$ with eigenvalues $0 = λ_1 < λ_2 \leq \dots \leq λ_n$. One can also consider the same graph equipped with positive edge weights $w:E \rightarrow \mathbb{R}_{> 0}$ normalized to $\sum_{e \in E} w_e = |E|$ and the associated weighted Laplacian matrix $L_w$. We say that $G$ is conformally rigid if constant edge-weights maximize the second eigenvalue $λ_2(w)$ of $L_w$ over all $w$, and minimize $λ_n(w')$ of $L_{w'}$ over all $w'$, i.e., for all $w,w'$, $$ λ_2(w) \leq λ_2(1) \leq λ_n(1) \leq λ_n(w').$$ Conformal rigidity requires an extraordinary amount of symmetry 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 into 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.

Figures (13)

• Figure 1: Three conformally rigid graphs. Left: Hoffman graph. Middle: complement of the Shrikhande graph. Right: CNG 6B.
• Figure 2: If a graph is conformally rigid, then the the interval spanned by the spectrum of the Laplacian, for any assignment of weights (fixing the total sum), always contains $[\lambda_2(\mathbbm{1}), \lambda_n(\mathbbm{1})]$.
• Figure 3: Summary of our main results.
• Figure 4: Three conformally rigid graphs that are distance-regular but neither strongly-regular nor edge-transitive. Left: a bipartite $(0,2)-$graph. Middle: a generalized quadrangle graph. Right: Doob graph $D(1,1)$.
• Figure 5: Left: The triangular prism graph is Cayley and not conformally rigid. Right: a Cayley graph on $\mathbb{Z}_{18}$ (generated by $S=\left\{-5,-1,1,5\right\}$) that is conformally rigid and not edge-transitive.

Theorems & Definitions (40)

• Definition 1.1
• Proposition 1.2
• Proposition 1.3
• Theorem 1.4: Fan-Taussky-Todd fan
• Corollary 1.5
• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Example 2.4
• Proposition 2.5