Multiplicity of 1 in Laplacian Spectra of trees

TL;DR

This paper investigates the multiplicity of 1 in the Laplacian spectra of trees, establishing conditions for equality in Faria's inequality and providing combinatorial procedures for computation.

Contribution

It introduces new conditions and procedures for computing the multiplicity of 1 in Laplacian spectra of trees, and explores related spectral properties.

Findings

Faria's inequality becomes an equality for normal trees without degree-2 vertices.

A combinatorial procedure for computing the multiplicity of 1 is proposed.

An inequality and computation method for the multiplicity of 0 in adjacency spectra are introduced.

Abstract

In this paper, we interpret the multiplicity of 1 in Laplacian spectra of trees and prove that Faria's inequality turns to an equality in the case of normal trees which yields that in any tree without a vertex of degree 2, Faria equality holds and multiplicity of 1 in Laplacian spectrum will be equal to star degree of the the tree. As a result we introduce a combinatorial procedure for computing the multiplicity of 1. In the way to prove this results we will introduce many transformation on graphs which are invariant regarding to the multiplicity of 1. We also introduce an inequality for the multiplicity of 0 in adjacency spectrum of graphs and again introduce a procedure to compute this multiplicity in the case of trees.