On Q-spectral integral variation

Let G be a connected graph with two nonadjacent vertices and G′ be 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. Hence, the sum of the eigenvalues of Q′ is the sum of the eigenvalues of Q plus 2. Since none of the eigenvalues of Q can decrease if an edge is added to G, it is said that Q-spectral integral variation occurs when either only one Q-eigenvalue is increased by 2, or when two Q-eigenvalues are increased by 1 one each. In this article we give necessary and sufficient conditions for the occurrence of Q-spectral integral variation only in two places, as the first case never occurs.