Graffiti conjecture 284

Write d(v) for the degree of v, and define its dual degree d^*(v) as the average of the degrees of its neighbors. For a connected graph with girth at least 5, Graffiti 284 asserted the following inequality, where λ_min(D) is the least eigenvalue of the graph's distance matrix.

The report that prompted this entry identifies the Hoffman–Singleton graph as a counterexample. The calculation below independently verifies the implication; the original Graffiti-record attribution remains flagged for bibliographic audit.

The Hoffman–Singleton graph H is the standard strongly regular graph with parameters (50, 7, 0, 1). In particular, it has 50 vertices, is 7-regular, has diameter 2, and has girth 5. Regularity immediately gives the left side: every neighbor of every vertex has degree 7.

Let A be H's adjacency matrix, I the identity, and J the all-ones matrix. Because H has diameter 2, distinct vertices are at distance 1 precisely on edges and at distance 2 otherwise. Therefore its distance matrix is:

The adjacency spectrum of H is 7¹, 2²⁸, (−3)²¹. On the all-ones line, D has eigenvalue 2(50−1)−7=91. On the orthogonal complement, J vanishes, so an A-eigenvalue θ becomes −2−θ for D. Hence:

Thus λ_min(D)=−4 and the conjectured right side is 4, while the left side is 7. H has the required girth, so it is a valid witness — no numerical eigensolver or graph search is involved.

Verdict: false. The substance of the counterexample is independently checked here; see the references for the standard graph data and the original social report.

• Report on X (Justin Sun, 2026-07-23)
• Hoffman–Singleton graph — DistanceRegular.org (50 vertices, diameter 2, spectrum 7¹ 2²⁸ (−3)²¹)
• Rowlinson & Sciriha (2007), Some properties of the Hoffman-Singleton graph (adjacency spectrum)
• Ichiro Shimada, The graphs of Hoffman–Singleton, Higman–Sims (standard strongly-regular parameters (50,7,0,1))