Generalising a conjecture due to Bollobas and Nikiforov

Abstract

Let µ 1 ≥ . . . ≥ µ n denote the eigenvalues of a graph G with m edges and clique number ω(G). Nikiforov proved a spectral version of Turán's theorem thatand Bollobás and Nikiforov conjectured that forThis paper proposes the conjecture that for all graphs (µ 2 1 + µ 2 2 ) in this inequality can be replaced by the sum of the squares of the ω largest eigenvalues, provided they are positive. We provide experimental and theoretical evidence for this conjecture, and describe how the bound can be applied.

Counting substructures and eigenvalues I: triangles

“…. For other extensions of Nosal's theorem, see [6,12,19,24]. In 2007, Bollobás and Nikiforov [2] proved a number of relations between the number of cliques of a graph G and λ(G).…”