On the Inertia Index of a Mixed Graph in Terms of the Matching Number

A mixed graph G^(-) is obtained by orienting some edges of G, where G is the underlying graph of G^(-) . The positive inertia index, denoted by p~+( G), and the negative inertia index, denoted by n~-(G^(-) ), of a mixed graph G^(-) are the integers specifying the numbers of positive and negative eigenvalues of the Hermitian adjacent matrix of G^(-) , respectively. In this paper, the positive and negative inertia indices of the mixed unicyclic graphs are studied. Moreover, the upper and lower bounds of the positive and negative inertia indices of the mixed graphs are investigated, and the mixed graphs which attain the upper and lower bounds are characterized respectively.