On the Spectra of General Random Mixed Graphs

A mixed graph is a that can be obtained from simple undirected by replacing some of the edges arcs in precisely one two possible directions. The Hermitian adjacency matrix $G$ order $n$ $n \times n$ $H(G)=(h_{ij})$, where $h_{ij}=-h_{ji}= \boldsymbol{\mathrm{i}}$ (with $\boldsymbol{\mathrm{i}} =\sqrt{-1})$ if there exists an arc $v_i$ to $v_j$ (but no $v_i$), $h_{ij}=h_{ji}=1$ edge (and arcs) between and $v_j$, $h_{ij}= 0$ otherwise (if are neither joined nor arc). We study spectra normalized Laplacian general random graphs, i.e., which all chosen independently with different probabilities regarded as oppositely oriented joining same pair vertices). For our first main result, we derive new probability inequality apply it obtain upper bound on eigenvalues matrix. Our second result shows approximated closely related weighted expectation matrix, error bounds depending minimum expected degree underlying graph.