# On the Displacement of Eigenvalues when Removing a Twin Vertex

## TL;DR

This paper investigates how removing a twin vertex from a graph affects the eigenvalues of its adjacency matrix, providing formulas and estimates for the spectral displacement caused by such perturbations.

## Contribution

It derives a closed-form expression for the characteristic polynomial of a graph with twin vertices and estimates the spectral displacement resulting from vertex removal.

## Key findings

- Closed formula for the characteristic polynomial with twin vertices.
- Estimates for eigenvalue displacement after vertex removal.
- Analysis of spectral perturbations in graph adjacency matrices.

## Abstract

Twin vertices of a graph have the same open neighbourhood. If they are not adjacent, then they are called duplicates and contribute the eigenvalue zero to the adjacency matrix. Otherwise they are termed co-duplicates, when they contribute $-1$ as an eigenvalue of the adjacency matrix. On removing a twin vertex from a graph, the spectrum of the adjacency matrix does not only lose the eigenvalue $0$ or $-1$. The perturbation sends a rippling effect to the spectrum. The simple eigenvalues are displaced. We obtain a closed formula for the characteristic polynomial of a graph with twin vertices in terms of two polynomials associated with the perturbed graph. These are used to obtain estimates of the displacements in the spectrum caused by the perturbation.