On Hermitian Adjacency Matrices for Mixed Graphs

Abstract

We study the spectra of mixed graphs about its Hermitian adjacency matrix of the second kind (i.e. N-matrix) introduced by Mohar [1]. We extend some results and define one new Hermitian adjacency matrix, and the entry corresponding to an arc from u to v is equal to the k-th( or the third) root of unity, i.e. ω = cos( 2π k ) + i sin( 2π k ), k ≥ 3; the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. In this paper, we characterize the cospectrality conditions for mixed graphs with the same underlying graph. In section 4, we determine a sharp upper bound on the spectral radius of mixed graphs, and provide the corresponding extremal graphs.