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

Yong Lu; Ligong Wang; Qiannan Zhou

doi:10.1186/s13660-017-1329-8

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

Yong Lu, Ligong Wang, Qiannan Zhou

School of Mathematics and Statistics

Northwestern Polytechnical University Xian

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

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 (v_k, v_l) is an arc of M, (rh)kl=(rh)lk=1dkdl if v_kv_l 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.

Original language	English
Article number	54
Journal	Journal of Inequalities and Applications
Volume	2017
Issue number	1
DOIs	https://doi.org/10.1186/s13660-017-1329-8
State	Published - 1 Dec 2017

Keywords

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

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10.1186/s13660-017-1329-8

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title = "Hermitian-Randi{\'c} matrix and Hermitian-Randi{\'c} energy of mixed graphs",

abstract = "Let M be a mixed graph and H(M) be its Hermitian-adjacency matrix. If we add a Randi{\'c} 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-Randi{\'c} 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-Randi{\'c} matrix of a mixed graph. Furthermore, we give bounds on the Hermitian-Randi{\'c} energy of a general mixed graph. Finally, we give some results about the Hermitian-Randi{\'c} energy of mixed trees.",

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AU - Lu, Yong

AU - Wang, Ligong

AU - Zhou, Qiannan

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Y1 - 2017/12/1

N2 - 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.

AB - 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.

KW - Hermitian-adjacency matrix

KW - Hermitian-Randić energy

KW - Hermitian-Randić matrix

KW - mixed graph

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JO - Journal of Inequalities and Applications

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

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