Signless Laplacians of finite graphs

Abstract

We survey properties of spectra of signless Laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix. For regular graphs the whole existing theory of spectra of the adjacency matrix and of the Laplacian matrix transfers directly to the signless Laplacian, and so we consider arbitrary graphs with special emphasis on the non-regular case. The results which we survey (old and new) are of two types: (a) results obtained by applying to the signless Laplacian the same reasoning as for corresponding results concerning the adjacency matrix, (b) results obtained indirectly via line graphs. Among other things, we present eigenvalue bounds for several graph invariants, an interpretation of the coefficients of the characteristic polynomial, a theorem on powers of the signless Laplacian and some remarks on star complements

Eigenvalue bounds for the signless laplacian

“…The matrix D + A is called the signless Laplacian in [16], and it appears very rarely in published papers (see [5]), the paper [13] being almost the only relevant research paper published before 2003. Only recently has the signless Laplacian attracted the attention of researchers [4,8,11,10,16,26]. The present paper extends the survey [8] by providing further comments, proofs and conjectures.…”

“…Only recently has the signless Laplacian attracted the attention of researchers [4,8,11,10,16,26]. The present paper extends the survey [8] by providing further comments, proofs and conjectures. The new results include some bounds for the eigenvalues of D + A.…”

“…The following proposition has been proved in [8]. In virtue of (1), the signless Laplacian is a positive semi-definite matrix, i.e., all its eigenvalues are non-negative.…”

“…Section 2 elaborates results from the survey paper [8] which will be required later. Section 3 contains the list of conjectures on Q-eigenvalues which have been formulated using the system AGX.…”

On the log-concavity of Laplacian characteristic polynomials of graphs

“…Using the terminology and notation from [4], a spanning subgraph of G whose connected components are trees or odd unicyclic graphs is called a T U -subgraph of G. Suppose that a T U -subgraph H of G contains c odd unicyclic graphs and s trees such as T 1 , . .…”