Addressing Graph Products and Distance-Regular Graphs

TL;DR

This paper investigates the minimal address length needed to encode distances in various graph families, building on Graham and Pollak's address assignment method, with new results for Hamming and triangular graphs.

Contribution

It introduces methods to determine minimal address lengths for specific graph families, notably Hamming and triangular graphs, advancing understanding of graph distance encoding.

Findings

Derived exact address lengths for Hamming graphs.

Established a lower bound for triangular graphs.

Extended address assignment techniques to new graph classes.

Abstract

Graham and Pollak showed that the vertices of any connected graph $G$ can be assigned $t$-tuples with entries in $\{0, a, b\}$, called addresses, such that the distance in $G$ between any two vertices equals the number of positions in their addresses where one of the addresses equals $a$ and the other equals $b$. In this paper, we are interested in determining the minimum value of such $t$ for various families of graphs. We develop two ways to obtain this value for the Hamming graphs and present a lower bound for the triangular graphs.