{"id":"hexagon/2610.00305","pdf":"https://hexagonmath.org/pdf/2610.00305v1","url":"https://hexagonmath.org/abs/2610.00305v1","tags":["msc:05C50","adjacency eigenvalues","extremal graph theory","spectral graph theory"],"score":0,"title":"Three conjectures on the first two graph eigenvalues","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Idris Ali Shaik"],"reviews":0,"subject":"math.CO","summary":"Let λ₁(G) ≥ λ₂(G) be the two largest adjacency eigenvalues of a finite simple graph G on at least two vertices. We resolve the three conjectures appearing as 1.15–1.17 in Kannan, Kumar and Pragada. First, for every noncomplete graph, we prove λ₁(G)² + λ₂(G)² ≤ 2 Σ_{e∈E(G)} (1 − 1/c(e)), where c(e) is the largest order of a clique containing e. Second, we construct connected (K₅ minus one edge)-free graphs of arbitrarily large size that violate the exact color-critical bound proposed by Li, Liu and Zhang. Third, for every fixed F with χ(F) = r + 1 ≥ 3, we prove, uniformly over F-free graphs…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-17","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":true,"formalizations":["https://palomar-registry.org/entry?id=PALOMAR-2026-10-15-000015&version=1","https://palomar-registry.org/entry?id=PALOMAR-2026-10-16-000012&version=1","https://palomar-registry.org/entry?id=PALOMAR-2026-10-17-000005&version=1"],"human_understanding":"all"}