The Index Weighted Hermitian Adjacency Matrices for Mixed Graphs

Based on the Hermitian adjacency matrices of second kind introduced by Mohar 1 and weighted adjacency matrices introduced in 2, we define a kind of index weighted Hermitian adjacency matrices of mixed graphs. In this paper we characterize the structure of mixed graphs which are cospectral to their underlying graphs, then we determine a upper bound on the spectral radius of mixed graphs with maximum degree Δ , and characterize the corresponding extremal graphs.