On the characteristic polynomials and H-ranks of the weighted mixed graphs.

Let D G ˜ be an n -vertex weighted mixed graph with Hermitian-adjacency matrix H (D G ˜ ). The characteristic polynomial of the weighted mixed graph D G ˜ is defined as ϕ (D G ˜ , λ) = det ⁡ (λ I n − H (D G ˜ )) = ∑ r = 0 n α r λ n − r and the H -rank of D G ˜ , written as r k (D G ˜ ) , is the rank of H (D G ˜ ). In this paper, we begin by interpreting all the coefficients of the characteristic polynomial ϕ (D G ˜ , λ). Then we establish recurrences for the characteristic polynomial ϕ (D G ˜ , λ). By these obtained results, as a unified approach, some main results obtained in Hou and Lei (2011) [13] , Gong and Xu (2012) [10] , Liu and Li (2015) [18] can be deduced consequently. Furthermore, for a weighted mixed bipartite cactus graph D G ˜ , we give a necessary and sufficient condition for r k (D G ˜ ) = 2 t , where t is any integer no greater than the matching number of G. Finally, as an application of our results, we study the minimum H -rank problem, determining the minimum H -rank of bipartite cactus graphs. [ABSTRACT FROM AUTHOR]