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

On the distance matrices of the CP graphs