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

Abstract

Abstract Let M be a mixed graph and H ( M ) $H(M)$ 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 R H ( M ) = ( r h ) k l $R_{H}(M)=(r_{h})_{kl}$ of a mixed graph M, where ( r h ) k l = − ( r h ) l k = i d k d l $(r_{h})_{kl}=-(r_{h})_{lk}=\frac{\mathbf{i}}{\sqrt {d_{k}d_{l}}}$ ( i = − 1 $\mathbf{i}=\sqrt{-1}$ ) if ( v k , v l ) $(v_{k},v_{l})$ is an arc of M, ( r h ) k l = ( r h ) l k = 1 d k d l $(r_{h})_{kl}=(r_{h})_{lk}=\frac{1}{\sqrt{d_{k}d_{l}}}$ if v k v l $v_{k}v_{l}$ is an undirected edge of M, and ( r h ) k l = 0 $(r_{h})_{kl}=0$ otherwise. In this paper, firstly, we compute the characteristic polynomial of the Hermitian-Randić matrix of a mixed graph. Furthermore, we give bounds on the Hermitian-Randić energy of a general mixed graph. Finally, we give some results about the Hermitian-Randić energy of mixed trees.

Keywords

• mixed graph
• Hermitian-adjacency matrix
• Hermitian-Randić matrix
• Hermitian-Randić energy