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

摘要

Parking functions are well researched and interesting results are found inthe listed references and more. Some introductory results stemming fromapplication to degree sequences of simple connected graphs are provided in thispaper. 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 $d(v_i)\geq2\},$ of a simple connected graph $G$ is a parking function, is presented. Wealso introduce the concept of \emph{looping degree sequences} and the\emph{looping number}, $\xi(G)$. Four open problems are proposed as well.