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combinatorics / Spectral graph theory

The Laplacian Sn,nS_{n,n} conjecture

True: no simple graph on n≥2n \ge 2 vertices has Laplacian spectrum {0,1,…,n−1}\{0,1,\ldots,n-1\}. The paper closes the range left between the two previously settled regimes, n≤15n \le 15 and n≥6 649 688 933n \ge 6\,649\,688\,933.

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combinatoricsSep 22, 2026Significance 22/100Registry: unreviewed

The Laplacian Sn,nS_{n,n} conjecture

Prior state unknown→proved

True: no simple graph on n≥2n \ge 2 vertices has Laplacian spectrum {0,1,…,n−1}\{0,1,\ldots,n-1\}. The paper closes the range left between the two previously settled regimes, n≤15n \le 15 and n≥6 649 688 933n \ge 6\,649\,688\,933.

Source authenticatedThe Laplacian Sn,nS_{n,n} conjecture →
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True: no simple graph on n≥2n \ge 2 vertices has Laplacian spectrum {0,1,…,n−1}\{0,1,\ldots,n-1\}. The paper closes the range left between the two previously settled regimes, n≤15n \le 15 and n≥6 649 688 933n \ge 6\,649\,688\,933.

True: no simple graph on n≥2n \ge 2 vertices has Laplacian spectrum {0,1,…,n−1}\{0,1,\ldots,n-1\}. The paper closes the range left between the two previously settled regimes, n≤15n \le 15 and n≥6 649 688 933n \ge 6\,649\,688\,933.

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