Based on the inverse results of some special graph matrices, a new class of graph (cp-graph with block K₄) for the inverse matrix solving was put forward. It was compared with the inverse matrix of complete graph, and the appropriate solution method is obtained. Then, its determinant and inverse matrix are obtained, and they are the functions related to the number of blocks in the graph. The eigenvalues and corresponding multiplicity of the distance matrix are obtained by the elementary determinant transformation.