Abstract: The inverse M-matrix is a class of very important nonnegative matrices, which has been widely used in many fields such as biology and physics. Using graph theory to study the completion of inverse M-matrix is an important direction in the field of inverse M-matrices. A double cycle graph is a directed graph built from two simple directed cycles intersecting at any number of vertices in structure. In this paper, we discuss the inverse M-matrix completion problem for this class of graphs. The necessary and sufficient conditions are presented for partial matrices having inverse M-matrix completions in two different cases, whose associated graphs are double cycles. The conditions are as follows: when all vertices in the double cycle graph are specified, the cycle product of each cycle is less than the product of its diagonal elements; when the vertices in the double cycle graph include unspecified vertices, each cycle contains at least one unspecified vertex. Furthermore, we present the specific completion algorithm, and the effectiveness of the algorithms is demonstrated by a numerical example.

Key words: inverse M-matrix, completion, $k$ double cycle, marked directed graph