A Robust Bollobás-Nikiforov Bound for Graphs Close to Triangle-Free Graphs

The Bollobás-Nikiforov (BN) conjecture, in its clique-number form, asserts that every non-complete graph G of order n with m edges and clique number ω(G) satisfies\[λ_1(G)^2+λ_2(G)^2≤ 2(1-1{ω(G)})m,\]where λ₁(G)≥ λ₂(G)≥⋯≥λₙ(G) are the adjacency eigenvalues of G.We prove an unconditional perturbative form for graphs that can be made triangle-free by deleting few edges. Let η_(G) be the minimum number of edges whose deletion destroys all triangles of G. Every non-complete graph G on at least two vertices satisfies\[λ_1(G)^2+λ_2(G)^2≤ 2(1-1ω)m+C_ω η_{}(G),\]where ω=ω(G) and\[C_ω=2+2ω+4ω/ω-2 (ω≥3).\]For ω=1,2, no additive error is needed. Since C_ω≤ 44/3, a uniform error term (44/3) η_(G) is valid. The emphasis of the result is the explicit finite error that is linear in the edge-deletion distance from the triangle-free class. We also show that the vertex-disjoint triangle packing number cannot replace η_(G) in this perturbative framework, and we clarify the relation with bounds expressed in terms of the total number of triangles.