Average Degree in Graph Powers

TL;DR

This paper establishes lower bounds on the average degree of graph powers for connected graphs with certain degree and diameter conditions, providing bounds that are nearly optimal.

Contribution

It introduces new lower bounds on the average degree of graph powers for specific classes of graphs, extending understanding of graph power properties.

Findings

Lower bound of (7/3)d on G^4 for graphs with minimum degree d > 2 and |V(G)| > (8/3)d.

Lower bound of (2k+1)(d+1) - k(k+1)(d+1)^2/n - 1 on G^{3k+2} for d-regular graphs with diameter ≥ 3k+3.

Results are essentially tight, with the second bound being optimal even as n/d grows large.

Abstract

The kth power of a simple graph G, denoted G^k, is the graph with vertex set V(G) where two vertices are adjacent if they are within distance k in G. We are interested in finding lower bounds on the average degree of G^k. Here we prove that if G is connected with minimum degree d > 2 and |V(G)| > (8/3)d, then G^4 has average degree at least (7/3)d. We also prove that if G is a connected d-regular graph on n vertices with diameter at least 3k+3, then the average degree of G^{3k+2} is at least (2k+1)(d+1) - k(k+1) (d+1)^2/n - 1. Both of these results are shown to be essentially best possible; the second is best possible even when n/d is arbitrarily large.