Trees with Four and Five Distinct Signless Laplacian Eigenvalues

Fatemeh Taghvaee, Gholam Hossein Fath-Tabar


‎‎Let $G$ be a simple graph with vertex set $V(G)=\{v_1‎, ‎v_2‎, ‎\cdots‎, ‎v_n\}$ ‎and‎

‎edge 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 $q_1$‎, ‎$q_2$‎, ‎$\cdots$‎, ‎$q_n$ in a graph with $n$ vertices‎.

‎In this paper we characterize all trees with four and five distinct signless Laplacian ‎eigenvalues.‎‎‎


‎Tree‎; ‎eigenvalue‎; ‎signless Laplacian matrix‎; ‎semi-edge walk

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