Characteristics polynomial of normalized Laplacian for trees

TL;DR

This paper derives the characteristic polynomial of the normalized Laplacian for trees, expressing coefficients via higher order Randić indices, and explores spectral properties of specific tree classes.

Contribution

It provides explicit formulas for the characteristic polynomial and Randić indices for starlike and double-starlike trees, and proves isomorphism of cospectral trees with same diameter.

Findings

Coefficients of the polynomial depend on the tree's structure.

Explicit formulas for Randić indices of specific trees.

Cospectral trees with same diameter are isomorphic.

Abstract

Here, we find the characteristics polynomial of normalized Laplacian of a tree. The coefficients of this polynomial are expressed by the higher order general Randi\'c indices for matching, whose values depend on the structure of the tree. We also find the expression of these indices for starlike tree and a double-starlike tree, $H_m(p,q)$. Moreover, we show that two cospectral $H_m(p,q)$ of the same diameter are isomorphic.