Hermitian-Randić matrix and Hermitian-Randić energy of mixed graphs

摘要

Let M be a mixed graph and H(M) be its Hermitian-adjacency matrix. If we add a Randić weight to every edge and arc in M, 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 RH(M) = (rh)kl of a mixed graph M, where (rh)kl=−(rh)lk=idkdl (i=−1) if (vk, vl) is an arc of M, (rh)kl=(rh)lk=1dkdl if vkvl is an undirected edge of M, and (rh)kl = 0 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.