On connected simple graphs and their degree sequences

TL;DR

This paper establishes precise conditions for when a degree sequence corresponds to a connected simple graph, introduces a matrix partitioning connected graphs, and relates edge count to graph connectedness.

Contribution

It provides necessary and sufficient conditions for degree sequences to be realizable as connected graphs and introduces a matrix to categorize connected graphs.

Findings

Characterization of degree sequences for connected graphs

Conditions for sequences to be necessarily connected

A matrix partitioning connected graphs

Abstract

This note describes necessary and sufficient conditions for a sequence of positive integers to be the degree sequence of a connected simple graph. Conditions are also given under which a sequence is necessarily connected i.e. the sequence can only be realised as a connected graph. A matrix is introduced whose non-empty entries partition the set of connected graphs. The note concludes with a result relating the number of edges in a simple graph to the connectedness of the graph.