A Matter of Degree

• Published in: SIAM journal on discrete mathematics Vol. 2; no. 4; pp. 456 - 466
• Main Authors: Chartrand, G., Hevia, H., Oellermann, O., Saba, F., Schwenk, A.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.11.1989
• Subjects: Applied mathematics
• ISSN: 0895-4801; 1095-7146
• DOI: 10.1137/0402040

The concepts of $n$th degrees and $n$th-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 $n$th 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 $n$th-order odd vertex of $G$ is a vertex adjacent to an odd number of $(n - 1)$st-order odd vertices. The number of $n$th-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}$$n$th-order odd vertices for every positive integer $n$. Generalized odd vertex sequences are characterized. Relationships between the $n$th 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.