Completely Non-negative Matrix, The P-matrix, N <sub> 0 </ Sub> <sup> 1 </ Sup> - Matrix Completed

Special matrices play an important role in matrix analysis and matrix computation and have wide applications in computational mathematics, applied mathematics, economics, physics, biology and etc. The special matrices have good properties so that the structure and property of special matrices are studied by applying graph theory and method. The relationship between graph theory and matrix theory is very close and graph theory used to discuss the structure and property of special matrices results in direct and compact effect. The use of graph theory to study matrices and the use of matrices to study graphs have complementary relationship. The use of graph theory to study matrices is an important fact. In this paper, matrix completion problems for patterns of certain types of matrices, such as totally nonnegative (TN) matrix, P -matrix and N₀¹-matrix, have been studied using the special graph theory.1. We have studied the k-subdirect sum problem of strictly diagonally dominant P -matrices and obtained that k-subdirect sum problem of k ( k≥2) strictly diagonally dominant P -matrices is a strictly diagonally dominant P -matrix. And we have shown the subdirect sum of two TN-matrices under some conditions is a TN-matrix and partial TN-matrix completion problem for monotonically labeled 2-chordal graph is studied.2. Based on the partial P -matrix completion problem for 1-chordal graph and 1-chordal block graph, we have obtained the partial positive P -matrices for k -chordal graph and for k -chordal block graph can be completed under some conditions. In addition, completion problem with zeros in the inverse completion are studied and the algorithms of partial P -matrix completion problem for k -chordal graph and k -chordal block graph are derived according to the structure of k -chordal graph. Last we give numerical example.3. Based on the partial N-matrix completion problem for 1-chordal graph and k-cycle, we have shown the problem completion of partial positionally symmetric N₀¹-matrix for k -cycle and 1-chordal graph.