Refined inertias of strongly connected orientations of the Petersen graph

Abstract

The 18 non-isomorphic strongly connected orientations of the Petersen graph give rise to matrix patterns in which nonzero entries can be taken to be strictly positive, of arbitrary sign, or of fixed sign. The allowed refined inertias, in which the number of zero eigenvalues is split from others on the imaginary axis, are considered for each such pattern. Each nonnegative pattern is shown to have unique refined inertia determined by the number of required zero eigenvalues. For zero–nonzero patterns, a complete list of allowed refined inertias is given for each orientation. One particular sign pattern is presented that allows only two distinct refined inertias out of a possible 161 for a sign pattern of order 10.