Maxima of the Q-index: forbidden even cycles

الأصل · EN

Let G be a graph of order n and let q(G) be the largest eigenvalue of the signless Laplacian of G. Let Sₙ,ₖ be the graph obtained by joining each vertex of a complete graph of order k to each vertex of an independent set of order n-k; and let Sₙ,ₖ⁺ be the graph obtained by adding an edge to Sₙ,ₖ. It is shown that if k≥2, n≥400k², and G is a graph of order n, with no cycle of length 2k+2, then q(G) <q(Sₙ,ₖ⁺), unless G=Sₙ,ₖ⁺. This result completes the proof of a conjecture of de Freitas, Nikiforov and Patuzzi.