On Lichnerowicz sharp distance-regular graphs

The first non-zero Laplacian eigenvalue $λ_1$ of a finite graph is bounded below by its minimum Lin–Lu–Yau curvature $κ$. This is a discrete analogue of the classical Lichnerowicz Theorem. A graph with $λ_1=κ$ is called Lichnerowicz sharp. In this note, we completely classify all Lichnerowicz sharp distance-regular graphs. Our result substantially strengthens the corresponding classification by Cushing, Kamtue, Koolen, Liu, Münch, and Peyerimhoff (Adv. Math. 2020), which required an extra spectral condition. As a key preparatory step, we provide a classification of all amply regular Terwilliger graphs with positive Lin-Lu-Yau curvature, a result that is interesting of its own right.

💡 Research Summary

The paper investigates the relationship between the first non‑zero eigenvalue λ₁ of the normalized Laplacian of a finite graph and its minimum Lin–Lu–Yau (LLY) curvature κ, an inequality λ₁ ≥ min_{xy∈E}κ(x,y) that mirrors the classical Lichnerowicz theorem in Riemannian geometry. A graph attaining equality, λ₁ = κ, is termed Lichnerowicz‑sharp. While earlier work (Cushing, Kamtue, Koolen, Liu, Münch, Peyerimhoff, Adv. Math. 2020) classified Lichnerowicz‑sharp distance‑regular graphs under the extra spectral condition θ₁ = b₁ − 1 (θ₁ being the second largest adjacency eigenvalue), the present note removes this restriction and provides a complete classification.

The authors’ strategy proceeds in two main phases. First, they classify all amply regular Terwilliger graphs with positive LLY curvature. An amply regular graph is a d‑regular graph on n vertices with parameters (n,d,α,β) where any two adjacent vertices share exactly α common neighbours and any two vertices at distance two share exactly β common neighbours. A Terwilliger graph is an amply regular graph in which the common‑neighbour subgraph of any distance‑two pair is a clique (equivalently, the graph contains no induced quadrangles). Using optimal transport arguments they prove Lemma 3.1, which gives a universal upper bound κ(x,y) ≤ (2α + 3 − d)/d for any edge xy. This bound forces strong restrictions on (α,β). In particular, when β = 1 the only possibilities are the pentagon, the icosahedron, and the Petersen line graph; when β > 1 the graph must arise as a line graph of a regular graph of girth at least five. Lemma 3.2 shows that if the underlying regular graph has diameter ≥ 3 then some edge of its line graph has non‑positive curvature, so a positive‑curvature line graph can only come from a regular graph of diameter two, i.e., a strongly regular graph. Combining these observations with known classifications of strongly regular graphs (Seidel’s list, the Hoffman–Singleton theorem, and the open case (3250,57,0,1)) yields Theorem 1.5: the only amply regular Terwilliger graphs with κ > 0 are the pentagon, the icosahedron, the line graph of the Petersen graph, the line graph of the Hoffman–Singleton graph, and the line graph of a (hypothetical) strongly regular graph with parameters (3250,57,0,1).

The second phase translates these curvature constraints into the language of distance‑regular graphs. Any non‑complete distance‑regular graph is automatically amply regular with parameters (n,b₀,b₀ − b₁ − 1,c₂). The authors invoke classical results (Bannai–Ito, Brouwer–Cohen–Neumaier) concerning the second largest eigenvalue θ₁ and the intersection numbers. Theorem 2.11 shows that if b⁺ = b₁/(θ₁+1) < 1 then either β = 1 or the graph is the icosahedron. Theorem 2.13 (the “θ₁ = b₁ − 1” classification) lists all possibilities for β: β = 1 (pentagon), β = 2 (Hamming, Doob, locally Petersen), β = 4 (Johnson), β = 6 (demicube), β = 10 (Gosset), together with the strongly regular case of smallest eigenvalue −2. Theorem 2.14 further restricts locally Petersen graphs to three known examples. By combining these structural constraints with the curvature bound κ ≥ λ₁ (the Lichnerowicz inequality) and the explicit curvature formula for regular graphs (Lemma 2.3), the authors deduce that the only distance‑regular graphs satisfying λ₁ = κ are:

• Cocktail‑party graphs CP(n) for n ≥ 2,
• Hamming graphs H(d,n) for d ≥ 1, n ≥ 2,
• Johnson graphs J(n,k) for 1 ≤ k ≤ n − 1,
• Demicubes Qₙ(2) for n ≥ 2,
• The Schläfli graph,
• The Gosset graph.

These six families coincide exactly with the list obtained in the earlier work, confirming that the extra spectral hypothesis θ₁ = b₁ − 1 is unnecessary. The paper concludes by highlighting the broader significance of classifying graphs with positive LLY curvature under various combinatorial restrictions (C₄‑free, outerplanar, Halin, planar with minimum degree three) and noting the longstanding open problem of the existence of a strongly regular graph with parameters (3250,57,0,1); its line graph would provide a new Lichnerowicz‑sharp example if it exists.

Overall, the work delivers a clean, curvature‑driven proof that fully resolves the classification of Lichnerowicz‑sharp distance‑regular graphs, while also advancing the understanding of curvature in highly symmetric graph families.