The doubly negative matrix completion problem

An n × n matrix over the field of real numbers is a doubly negative matrix if it is symmetric, negative definite and entry-wise negative. In this paper, we are interested in the doubly negative matrix completion problem, that is when does a partial matrix have a doubly negative matrix completion. In general, we cannot guarantee the existence of such a completion. In this paper, we prove that every partial doubly negative matrix whose associated graph is a p-chordal graph G has a doubly negative matrix completion if and only if p = 1. Furthermore, the question of completability of partial doubly negative matrices whose associated graphs are cycles is addressed.