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

Abstract

Let M be a mixed graph and 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 of a mixed graph M, where () if is an arc of M, if is an undirected edge of M, and 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.

On First Hermitian-Zagreb Matrix and Hermitian-Zagreb Energy

“…A real spanning elementary subgraph of a mixed graph M is an elementary subgraph such that it contains all vertices of M and all its mixed cycles are real. Now we will give two results which are similar to those in [11]. The proofs of these results are avoided because they are similar to the ones done by Lu et al, in [11].…”

“…Now we will give two results which are similar to those in [11]. The proofs of these results are avoided because they are similar to the ones done by Lu et al, in [11].…”

“…Recently in 2018, Rad et al, communicated some results on Zagreb energy and Zagreb Estrada index of graphs [12]. In these studies researchers consider only undirected graphs, but in last few years some attempts are also reported for oriented as well as mixed graphs for Randi ́ index [4], [11].…”

Some Less Familiar Properties of Randić Index

“…The most straightforward was to conceive a "general Randić energy", [45] containing a variable exponent α. Of the variety of modified Randić-type energies, we mention here the following: Hermite-Randić energy [46] Randić incidence energy [47] minimum dominating Randić energy [48] minimum equitable dominating Randić energy [49,50] Randić color energy [51] minimum covering Randić energy [52,53] Laplacian minimum covering Randić energy [54] skew Randić energy [55] Randić-type additive connectivity energy [56]…”