Bordering of Symmetric Matrices and an Application to the Minimum Number of Distinct Eigenvalues for the Join of Graphs

TL;DR

This paper investigates how bordering matrices affects the minimum number of distinct eigenvalues in graph joins, providing new results for specific graph operations and their spectral properties.

Contribution

It introduces bordering techniques to analyze eigenvalue changes in graph joins, advancing understanding of spectral graph theory.

Findings

Resolved the minimum eigenvalues problem for the join of a connected graph with a path.

Developed bordering methods to control eigenvalue variations during graph operations.

Provided numerous results on eigenvalue behavior in graph joins.

Abstract

An important facet of the inverse eigenvalue problem for graphs is to determine the minimum number of distinct eigenvalues of a particular graph. We resolve this question for the join of a connected graph with a path. We then focus on bordering a matrix and attempt to control the change in the number of distinct eigenvalues induced by this operation. By applying bordering techniques to the join of graphs, we obtain numerous results on the nature of the minimum number of distinct eigenvalues as vertices are joined to a fixed graph.