Enumeration of cospectral graphs

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.…”

The maximum clique and the signless Laplacian eigenvalues

“…The matrix L = D − A is known as the Laplacian of G and has been studied extensively in literature (see, e.g., [1]). The matrix Q = D + A is called the signless Laplacian in [4] and appears very rarely in published papers (see [1]), the paper [2] being one of the very few research papers concerning this matrix. Since the signless Laplacian is a positive semi-definite matrix, all its eigenvalues are non-negative.…”

Some properties of the distance Laplacian eigenvalues of a graph

“…The first infinite family of pairs of A-cospectral mates trees was constructed by Schwenk [23], who also proved that asymptotically every tree has a mate. The L-cospectrality is studied in [13], [15], [16], [20], [21], [24]. The smallest L-cospectral graphs, with respect to the order, contain 6 vertices, and are given in Figure 3.…”