## Abstract

A mixed graph is a graph that can be obtained from a simple undirected graph by replacing some of the edges by arcs in precisely one of the two possible directions. The Hermitian adjacency matrix of a mixed graph G of order n is the n×n matrix H(G)=(h_{ij}), where h_{ij}=−h_{ji}=i (with i=−1) if there exists an arc from v_{i} to v_{j} (but no arc from v_{j} to v_{i}), h_{ij}=h_{ji}=1 if there exists an edge (and no arcs) between v_{i} and v_{j}, and h_{ij}=h_{ji}=0 otherwise (if v_{i} and v_{j} are neither joined by an edge nor by an arc). Let λ_{1}(G),λ_{2}(G),…,λ_{n}(G) be eigenvalues of H(G). The k-th Hermitian spectral moment of G is defined as s_{k}(H(G))=∑_{i=1}^{n}λ_{i}^{k}(G), where k≥0 is an integer. In this paper, we deal with the asymptotic behavior of the spectrum of the Hermitian adjacency matrix of random mixed graphs. We will present and prove a separation result between the largest and remaining eigenvalues of the Hermitian adjacency matrix, and as an application, we estimate the Hermitian spectral moments of random mixed graphs.

Original language | English |
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Pages (from-to) | 320-338 |

Number of pages | 19 |

Journal | Linear algebra and its applications |

Volume | 653 |

DOIs | |

Publication status | Published - 15 Nov 2022 |

## Keywords

- Hermitian spectral moment
- Random Hermitian adjacency matrix
- Random mixed graphs
- Spectrum
- UT-Hybrid-D