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.