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

2016/10/31 by Lu, Yong, Wang, Ligong, Zhou, Qiannan · 1 citation
#05C07 #05C31 #05C50 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1610.09783

Abstract

Let M be a mixed graph and H(M) be its Hermitian-adjacency matrix. If we add every edge and arc in M a Randić weight, 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 R_H(M)=(r_h)_kl of a mixed graph M, where (r_h)_kl=-(r_h)_lk=\fraci√d_kd_l (i=√(-1)) if (v_k,v_l) is an arc of M, (r_h)_kl=(r_h)_lk=\frac1√d_kd_l if v_kv_l is an undirected edge of M, and (r_h)_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 to the Hermitian-Randić energy of a general mixed graph. Finally, we give some results about the Hermitian-Randić energy of mixed trees.

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