Towards a Spectral Theory of Graphs Based on the Signless Laplacian, Iii

This is the third part of our work with a common title. The first [11] and the second part [12] will be also referred in the sequel as Part I and Part II, respectively. This third part was not planned at the beginning, but a lot of recently published papers on the signless Laplacian eigenvalues of graphs and some observations of ours justify its preparation. By a spectral graph theory we understand, in an informal sense, a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M–theory. Hence, there are several spectral graph theories (for example, those based on the adjacency matrix, the Laplacian, etc.). In that sense, the title “Towards a spectral theory of graphs based on the signless Laplacian” indicates the intention to build such a spectral graph theory (the one which uses the signless Laplacian without explicit involvement of other graph matrices). Recall that, given a graph, the matrix Q = D + A is called the signless Laplacian, where A is the adjacency matrix and D is the diagonal matrix of vertex degrees. In fact, we outlined in [11], [12] a new spectral theory of graphs (based on the signless Laplacian Q). We shall call this theory the Q–theory.