On the distance matrices of the CP graphs

TL;DR

This paper introduces CP graphs, a new class of graphs with structure-independent distance properties, and explores their applications to the addressing problem, expanding understanding of graph invariants.

Contribution

It defines CP graphs and proves their distance determinant and inertia are structure-independent, also showing these properties hold when attached to other graphs.

Findings

Distance determinant and inertia are structure-independent for CP graphs.

Attaching CP graphs preserves these distance properties.

Applications to the Graham-Pollak addressing problem are provided.

Abstract

This paper introduces a new class of graphs, the CP graphs, and shows that their distance determinant and distance inertia are independent of their structures. The CP graphs include the family of linear $2$-trees. When a graph is attached with a CP graph, it is shown that the distance determinant and the distance inertia are also independent of the structure of the CP graph. Applications to the addressing problem proposed by Graham and Pollak in 1971 are given.