Taiwanese Journal of Mathematics

Normalized Laplacian Eigenvalues and Energy of Trees

Kinkar Ch. Das and Shaowei Sun

Full-text: Open access

Abstract

Let $G$ be a graph with vertex set $V(G) = \{v_1, v_2, \ldots, v_n\}$ and edge set $E(G)$. For any vertex $v_i \in V(G)$, let $d_i$ denote the degree of $v_i$. The normalized Laplacian matrix of the graph $G$ is the matrix $\mathcal{L} = (\mathcal{L}_{ij})$ given by\[\mathcal{L}_{ij} =\begin{cases}1 &\textrm{if $i = j$ and $d_{i} \neq 0$} \\-\frac{1}{\sqrt{d_{i} \,d_{j}}} &\textrm{if $v_i v_j \in E(G)$} \\0 &\textrm{otherwise}.\end{cases}\] In this paper, we obtain some bounds on the second smallest normalized Laplacian eigenvalue of tree $T$ in terms of graph parameters and characterize the extremal trees. Utilizing these results we present some lower bounds on the normalized Laplacian energy (or Randić energy) of tree $T$ and characterize trees for which the bound is attained.

Article information

Source
Taiwanese J. Math., Volume 20, Number 3 (2016), 491-507.

Dates
First available in Project Euclid: 1 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498874460

Digital Object Identifier
doi:10.11650/tjm.20.2016.6668

Mathematical Reviews number (MathSciNet)
MR3511991

Zentralblatt MATH identifier
1357.05077

Subjects
Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)

Keywords
tree normalized Laplacian matrix normalized Laplacian eigenvalues normalized Laplacian energy

Citation

Das, Kinkar Ch.; Sun, Shaowei. Normalized Laplacian Eigenvalues and Energy of Trees. Taiwanese J. Math. 20 (2016), no. 3, 491--507. doi:10.11650/tjm.20.2016.6668. https://projecteuclid.org/euclid.twjm/1498874460


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