AbstractThe path layer matrix (or path degree sequence) of a graphGcontains quantitative information about all paths inG. The entry (i,j) in this matrix is the number of simple paths inGhaving initial vertexvand lengthj.It was known that there are cubic graphs on 90 vertices having the same path layer matrix (Dobrynin, 1990). A new upper bound for the least order of cubic graphs with the same path layer matrix is presented. © 1993 John Wiley&Sons, Inc