Trees with Four and Five Distinct Signless Laplacian Eigenvalues
Abstract
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.
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References
Ayoobi, F., Omidi, G.R. and Tayfeh-Rezaie B., A note on graphs whose signless Laplacian
has three distinct eigenvalues, Linear Multilinear Algebra, 59 (2011), 701-706.
Barik, S., Pati, S. and Sarma, B.K., The spectrum of the corona of two graphs, SIAM J.
Discrete Math., 21 (2007), 47-56.
Biggs, N., Algebraic Graph Theory, Cambridge Univ, Press, Cambridge 1974.
Bridges, W.G. and Mena R.A., Multiplicative cones - a family of three eigenvalue graphs,
Aequationes Math., 22 (1981), 208-214.
Brouwer, A.E. and W.H. Haemers, spectra of graph, Springer, New York, etc. 2012.
Chuang, H. and Omidi, G.R., Graphs with three distinct eigenvalues and largest eigenvalue
less than 8, Linear Algebra Appl., 430 (2009), 2053-2062.
Cvetkovi´c, D., New theorems for signless Laplacian eigenvalues, Bull. Cl. Sci. Math. Nat.
Sci. Math., 137 (2008), 131-146.
Cvetkovi´c, D., Rowlinson, P. and Simi´c, S.K., Signless Laplacians of finite graphs, Linear
Algebra Appl., 423 (2007), 155-171.
Cvetkovi´c, D. and Simi´c, S.K., Towards a spectral theory of graphs based on the signless
Laplacian, II, Linear Algebra Appl., 432 (2010), 2257-2272.
Ellahi, H.R., Fath-Tabar, G.H., Gholami, A. and Nasiri, R., On maximum signless Laplacian
Estrada index of graphs with given parameters, Ars Math. Contemp., 11 (2016), 381-389.
Fan, Y.Z., Tam, B.S. and Zhou, J., Maximizing spectral radius of unoriented Laplacian
matrix over bicyclic graphs of a given order, Linear Multilinear Algebra, 56 (2008), 381-397.
Mohammadian, A. and Tayfeh-Rezaie, B., Graphs with four distinct Laplacian eigenvalues,
J. Algebraic Combin., 34 (2011), 671-682.
Taghvaee, F. and Fath-Tabar, G.H., Signless Laplacian spectral moments of graphs and
ordering some graphs with respect to them, Alg. Struc. Appl., 1 (2014), 133-141.
Taghvaee, F. and Fath-Tabar, G.H., On the skew spectral moments of graphs, Trans. Comb.,
(2017), 47-54.
Taghvaee, F. and Fath-Tabar, G.H., Relationship between coefficients of characteristic polynomial and matching polynomial of regular graphs and its applications, Iranian J. Math. Chem., 8 (2017), 7-23.
Tavakoli, M., Rahbarnia, F. and Ashrafi, A.R., Studying the corona product of graphs under
some graph invariants, Trans. Comb., 3 (2014), 43-49.
van Dam, E.R., Regular graphs with four eigenvalues, Linear Algebra Appl., 226/228 (1995),
-162.
van Dam, E.R., Graphs with few eigenvalues, an interplay between combinatorics and algebra,
Center Dissertation Series 20, Thesis, Tilburg University 1996.
Wang, J., Shen, Y. and Huang, Q., Notes on graphs with least eigenvalue at least −2,
Electron. J. Linear Algebra, 23 (2012), 387-396.
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