Number of walks and degree powers in a graph

TL;DR

This paper establishes an upper bound on the total number of k-walks in a graph based on the sum of the k-th powers of vertex degrees, linking walk counts to degree distributions.

Contribution

It proves that the total number of k-walks in a graph is bounded above by the sum of the k-th powers of its vertex degrees, providing a new analytical relationship.

Findings

Number of k-walks is bounded by degree power sums.

Provides a mathematical link between walks and degree distribution.

Offers a new perspective for analyzing graph structure.

Abstract

This note deals with the relationship between the total number of $k$-walks in a graph, and the sum of the $k$-th powers of its vertex degrees. In particular, it is shown that the the number of all $k$-walks is upper bounded by the sum of the $k$-th powers of the degrees.