  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.
