Research output per year
Research output per year
Dan Hu, Hajo Broersma^{*}, Jiangyou Hou, Shenggui Zhang
Research output: Contribution to journal › Article › Academic › peer-review
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} = 0 otherwise (if v_{i} and v_{j} are neither joined by an edge nor by an arc). We study the spectra of the Hermitian adjacency matrix and the normalized Hermitian Laplacian matrix of general random mixed graphs, i.e., in which all arcs are chosen independently with different probabilities (and an edge is regarded as two oppositely oriented arcs joining the same pair of vertices). For our first main result, we derive a new probability inequality and apply it to obtain an upper bound on the eigenvalues of the Hermitian adjacency matrix. Our second main result shows that the eigenvalues of the normalized Hermitian Lapla-cian matrix can be approximated by the eigenvalues of a closely related weighted expectation matrix, with error bounds depending on the minimum expected degree of the underlying undirected graph.
Original language | English |
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Article number | P1.3 |
Number of pages | 23 |
Journal | The Electronic journal of combinatorics |
Volume | 28 |
Issue number | 1 |
DOIs | |
Publication status | Published - 15 Jan 2021 |
Research output: Contribution to conference › Paper › peer-review