## Taiwanese Journal of Mathematics

### Normalized Laplacian Eigenvalues and Energy of Trees

#### 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

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.)

#### 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

#### References

• J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., New York, 1976.
• R. O. Braga, R. R. Del-Vechio, V. M. Rodrigues and V. Trevisan, Trees with $4$ or $5$ distinct normalized Laplacian eigenvalues, Linear Algebra Appl. 471 (2015), 615–635.
• M. Cavers, The Normalized Laplacian Matrix and General Randić Index of Graphs, Ph.D. dissertation, University of Regina, 2010.
• M. Cavers, S. Fallat and S. Kirkland, On the normalized Laplacian energy and general Randić index $R_{-1}$ of graphs, Linear Algebra Appl. 433 (2010), no. 1, 172–190.
• G. Chen, G. Davis, F. Hall, Z. Li, K. Patel and M. Stewart, An interlacing result on normalized Laplacians, SIAM J. Discrete Math. 18 (2004), no. 2, 353–361.
• F. R. K. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 92, 1997.
• K. Ch. Das, A. D. Güngör and Ş. B. Bozkurt, On the normalized Laplacian eigenvalues of graphs, Ars Combin. 118 (2015), 143–154.
• K. Ch. Das and S. Sorgun, On Randić energy of graphs, MATCH Commun. Math. Comput. Chem. 72 (2014), no. 1, 227–238.
• K. Ch. Das, S. Sorgun and I. Gutman, On Randić energy, MATCH Commun. Math. Comput. Chem. 73 (2015), no. 1, 81–92.
• I. Gutman, B. Furtula and Ş. B. Bozkurt, On Randić energy, Linear Algebra Appl. 442 (2014), 50–57.
• J. Li, J.-M. Guo and W. C. Shiu, A note on Randić energy, MATCH Commun. Math. Comput. Chem. 74 (2015), no. 2, 389–398.
• J. Li, J.-M. Guo, W. C. Shiu and A. Chang, An edge-separating theorem on the second smallest normalized Laplacian eigenvalue of a graph and its applications, Discrete Appl. Math. 171 (2014), 104–115.
• ––––, Six classes of trees with largest normalized algebraic connectivity, Linear Algebra Appl. 452 (2014), 318–327.