Null Decomposition of Trees

TL;DR

This paper introduces the Null Decomposition of trees, revealing how the null space of a tree's adjacency matrix encodes structural properties and relates to key graph invariants like independence and matching numbers.

Contribution

The paper presents a novel Null Decomposition method for trees, linking null space properties to combinatorial parameters and maximum matchings, advancing structural understanding.

Findings

Null space of adjacency matrix encodes tree structure

Formulas for independence and matching numbers derived

Number of maximum matchings related to null decomposition

Abstract

Let $T$ be a tree, we show that the null space of the adjacency matrix of $T$ has relevant information about the structure of $T$. We introduce the Null Decomposition of trees, and use it in order to get formulas for independence number and matching number of a tree. We also prove that the number of maximum matchings in a tree is related to the null decomposition.