Graphs with at most one signless Laplacian eigenvalue exceeding three

Abstract

We determine all connected graphs with at most one signless Laplacian eigenvalue exceeding three.

On the Signless Laplacian Spectral Radius of Cacti

“…The characteristic polynomial of Q(Gn,k,p) was given in the proof of Lemma 4 in Ref. [8], from which the first part follows.…”

“…The study of the signless Laplacian spectral radius of graphs has received much attention. [2][3][4][5][6][7][8][9][10] A cactus is a connected graph in which every edge appears in at most one cycle, see, e.g. Ref.…”

Sharp upper bounds of $A_\alpha$-spectral radius of cacti with given pendant vertices

“…The bounds of signless Laplacian spectral radius and its hamiltonicity are studied by Zhou [24]. Lin and Zhou [13] obtained graphs with at most one signless Laplacian eigenvalue larger than three. In addition to the successful considerations of these spectral radius, A α -spectral radius is provided as a general version of adjacency and signless Laplacian radius, and this area would be challenging.…”

The Aα-Spectral Radii of Graphs with Given Connectivity

“…The research of (adjacency, signless Laplacian) spectral radius is an intriguing topic during past decades [3][4][5][6][7][8][9]. For instances, Lovász and J. Pelikán studied the spectral radius of trees [10].…”

“…Zhou [13] found bounds of signless Laplacian spectral radius and its hamiltonicity. Graphs having none or one signless Laplacian eigenvalue larger than three are obtained by Lin and Zhou [14]. At the same time, the maximal adjacency or signless Laplacian spectral radius have attracted many interests among the mathematical literature including algebra and graph theory.…”