The normalized Laplacian spectrum of subdivisions of a graph

TL;DR

This paper derives the normalized Laplacian spectra for iterated subdivisions of graphs, enabling precise calculations of various graph invariants related to structure and dynamics.

Contribution

It provides explicit spectra for subdivided graphs, linking spectral properties to important graph invariants like Kirchhoff index and spanning trees.

Findings

Exact spectra for subdivided graphs obtained.

Closed-form formulas for graph invariants derived.

Enhanced understanding of spectral properties in graph subdivisions.

Abstract

Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spectra of the normalized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact values of their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees.