Trees with Four and Five Distinct Signless Laplacian Eigenvalues - Fath-Tabar, Gholam Hossein; Taghvaee, Fatemeh (IndoMS, 2019)
‎‎Let G be a simple graph with vertex set V(G)={v_1‎, ‎v_2‎, ‎…‎, ‎v_n} ‎and‎‎edge set E(G)‎.‎The signless Laplacian matrix of G is the matrix ‎Q‎‎=‎D‎+‎A‎‎‎, ‎such that D is a diagonal ‎matrix‎%‎‎, ‎indexed by the vertex set of G where‎‎%‎D_ii is the degree of the vertex v_i ‎‎‎ and A is the adjacency matrix of G‎.‎%‎ where A_ij = 1 when there‎‎%‎‎is an edge from i to j in G and A_ij = 0 otherwise‎.‎The eigenvalues of Q is called the signless Laplacian eigenvalues of G and denoted by q_1‎, ‎q_2‎, ‎…‎, ‎q_n in a graph with n vertices‎.‎In this paper we characterize all trees with four and five distinct signless Laplacian ‎eigenvalues.‎‎‎