# On the Degree Sequences of Multigraphs with Edge Additions and Deletions

## TL;DR

This paper investigates how degree sequences of multigraphs change under edge additions and deletions, proposing conjectures for their characterization across various graph families.

## Contribution

It introduces conjectures for characterizing degree sequences of multigraphs after edge modifications, extending existing theories to new graph classes.

## Key findings

- Proposes conjectures for degree sequence characterizations
- Extends known results to threshold graphs and complete multigraphs
- Provides a framework for future proofs and validations

## Abstract

The degree sequence of a graph is a numerical method to characterize the properties of graphs. Generalized forms of degree sequences exist for complete graphs and complete graphs. Nikolopolus et al. characterized the number of spanning trees from edge deletions and edge additions. Instead of investigating the number of spanning trees of graphs that arise from edge additions and deletions, we sought to characterize degree sequences of such graphs. We conjecture a characterization for the degree sequence of the addition and edge deletion for many families of graphs including threshold graphs and complete multigraphs. Keywords: multigraphs, split graphs, degree sequence, threshold graph, Havel-Hakimi, Ruch-Gutman, Edge Deletion