A Matter of Degree

Abstract

The concepts of nth degrees and nth-order odd vertices in graphs are introduced. The first degree of a vertex v in a graph G is the degree of v, while the nth degree ($n\geqq 2$) of $v $ is the sum of the $(n - 1)$st degrees of the vertices adjacent to $v $ in G. By a first-order odd vertex in a graph G is meant an (ordinary) odd vertex in G, while for $n\geqq 2$, an nth-order odd vertex of G is a vertex adjacent to an odd number of $(n - 1)$st-order odd vertices. The number of nth-order odd vertices, $n = 1,2, \cdots $, is investigated. A sequence $s_{1}, s_{2}, \cdots ,s_n , \cdots $ of integers is defined to be a generalized odd vertex sequence if there exists a graph G containing exactly $s_{n}$nth-order odd vertices for every positive integer n. Generalized odd vertex sequences are characterized. Relationships between the nth degrees of the vertices of a graph G and the walks of length n in G are described. The analogous problem for digraphs is also discussed.