Mixed graphs whose Hermitian adjacency matrices of the second kind have the smallest eigenvalue greater than [formula omitted

Abstract

Mixed graphs unify simple graphs and oriented graphs naturally. The Hermitian adjacency matrix of a mixed graph for the second kind (N-matrix for short) was recently introduced by Mohar [13]. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the principal sixth root of unity ω=1+i32 (and its symmetric entry is ω‾=1−i32); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. In this paper, we completely characterize the connected mixed graphs whose N-matrices have the smallest eigenvalue greater than −32. It may be considered as the continuance of the work of J.H. Smith [18] who considered the classical problem of characterizing the graphs whose eigenvalues lie in a given interval.