The Derivative Degree Sequences of Finite Simple Connected Graphs are Parking Functions

TL;DR

This paper explores the relationship between degree sequences of connected graphs and parking functions, introducing new concepts like looping degree sequences and proposing open problems in the area.

Contribution

It proves that derivative degree sequences of connected graphs are parking functions and introduces the concepts of looping degree sequences and looping number.

Findings

Derivative degree sequences are parking functions.

Introduces looping degree sequences and looping number.

Proposes four open problems in the field.

Abstract

Parking functions are well researched and interesting results are found in the listed references and more. Some introductory results stemming from application to degree sequences of simple connected graphs are provided in this paper. Amongst others, the result namely, that a derivative degree sequence, $d_d(G) \in \Bbb D_d(G)= \{(\lceil\frac{d(v_1}{\ell}\rceil, \lceil\frac{d(v_2)}{\ell}\rceil, \lceil\frac{d(v_3)}{\ell}\rceil, ..., \lceil\frac{d(v_n)}{\ell}\rceil| \ell = d(v_i), \forall i,$ with $d(v_i)\geq 2\},$ of a simple connected graph $G$ is a parking function, is presented. We also introduce the concept of \emph{looping degree sequences} and the \emph{looping number}, $\xi(G)$. Four open problems are proposed as well.