The α-Hermitian adjacency matrix Hα of a mixed graph X has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of the complex number α. This enables us to define an incidence matrix of mixed graphs. Consequently, we define a generalization of line graphs as well as a generalization of the signless Laplacian adjacency matrix of graphs. We then study the spectral properties of the gamma-signless Laplacian adjacency matrix of a mixed graph. Lastly, we characterize when the signless Laplacian adjacency matrix of a mixed graph is singular and give lower and upper bounds of number of arcs and digons in terms of largest and lowest eigenvalue of the signless Laplacian adjacency matrix.
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Alomari, Omar; Abudayah, Mohammad; and Ghanem, Manal
"The gamma-Signless Laplacian Adjacency Matrix of Mixed Graphs,"
Theory and Applications of Graphs: Vol. 10:
1, Article 11.
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol10/iss1/11