Hermitian-Randić matrix and Hermitian-Randić energy of mixed graphs.
LetMbe a mixed graph and [Formula: see text] be its Hermitian-adjacency matrix. If we add a Randić weight to every edge and arc inM, then we can get a new weighted Hermitian-adjacency matrix. What are the properties of this new matrix? Motivated by this, we define the Hermitian-Randić matrix [Formula: see text] of a mixed graphM, where [Formula: see text] ([Formula: see text]) if [Formula: see text] is an arc ofM, [Formula: see text] if [Formula: see text] is an undirected edge ofM, and [Formula: see text] otherwise. In this paper, firstly, we compute the characteristic polynomial of the Hermitian-Randić matrix of a mixed graph. Furthermore, we give bounds on the Hermitian-Randić energy of a general mixed graph. Finally, we give some results about the Hermitian-Randić energy of mixed trees.
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi'an, Shaanxi 710072 People's Republic of China.
Yong
Y
Lu
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi'an, Shaanxi 710072 People's Republic of China.
Ligong
L
Wang
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi'an, Shaanxi 710072 People's Republic of China.
Qiannan
Q
Zhou
Hermitian-Randić energy; Hermitian-Randić matrix; Hermitian-adjacency matrix; mixed graph