On Q-spectral integral variation

Let G be a graph with two non adjacent vertices and G′ the graph constructed from G by adding an edge between them. It is known that the trace of Q′ is 2 plus the trace of Q, where Q and Q′ are the signless Laplacian matrices of G and G′ respectively. So, the sum of the Q′-eigenvalues of G′ is the sum of the the Qeigenvalues of G plus two. It is said that Q-spectral integral variation occurs when either only one Q-eigenvalue is increased by two or two Q-eigenvalues are increased by 1 each one. In this article we present some conditions for the occurrence of Q-spectral integral variation under the addition of an edge to a graph G.