图的一种加权邻接矩阵谱半径和能量的界

兰州理工大学学报 ›› 2023, Vol. 49 ›› Issue (1): 144-151.

图的一种加权邻接矩阵谱半径和能量的界

卢鹏丽^*, 薛小燕

兰州理工大学计算机与通信学院, 甘肃兰州 730050

收稿日期:2021-12-21 出版日期:2023-02-28 发布日期:2023-03-21
通讯作者: 卢鹏丽(1973-),女,甘肃酒泉人,教授,博导. Email:lupengli88@163.com
基金资助:
国家自然科学基金(11361033,11861045,62162040)

Bounds for the spectral radius and energy of a kind of weighted adjacency matrix of graphs

LU Peng-li, XUE Xiao-yan

School of Computer and Communication, Lanzhou Univ. of Tech., Lanzhou 730050, China

Received:2021-12-21 Online:2023-02-28 Published:2023-03-21

摘要/Abstract

摘要： 图G的一种加权邻接矩阵记为A_db(G)=(a^db_ij)_n×n,若顶点v_i和顶点v_j相邻,则$a_{i j}^{d b}=\frac{d_{i}+d_{j}}{d_{i} d_{j}}$, 反之a^db_ij=0.给出图G的加权谱半径的上下界,并在此基础上给出加权谱半径的Nordhaus-Gaddum-type关系.得到了图G的加权能量的几个上下界,并在此基础上给出加权能量的Nordhaus-Gaddum-type关系.

关键词: 一种加权邻接矩阵, 加权谱半径, 加权能量, Nordhaus-Gaddum-type关系

Abstract: A kind of weighted adjacency matrix of graph G is denoted by A_db(G)=(a^db_ij)_n×n, where $a_{i j}^{d b}=\frac{d_{i}+d_{j}}{d_{i} d_{j}}$ if the vertices v_i and v_j are adjacent, and a^db_ij=0 otherwise. In this paper, some upper and lower bounds of the weighted spectral radius of graph G are first obtained, and then the Nordhaus-Gaddum-type relation for weighted spectral radius is given. In addition, several upper and lower bounds for the weighted energy are obtained, then, the Nordhaus-Gaddum-type results for the weighted energy are presented.

Key words: a kind of weighted adjacency matrix, weighted spectral radius, weighted energy, Nordhaus-Gaddum-type relation

中图分类号:

卢鹏丽, 薛小燕. 图的一种加权邻接矩阵谱半径和能量的界[J]. 兰州理工大学学报, 2023, 49(1): 144-151.

LU Peng-li, XUE Xiao-yan. Bounds for the spectral radius and energy of a kind of weighted adjacency matrix of graphs[J]. Journal of Lanzhou University of Technology, 2023, 49(1): 144-151.

参考文献

[1] YANG Y Q,XU L,HU C Y.Extended adjacency matrix indices and their applications [J].Journal of Chemical Information and Computer Sciences,1994,34(5):1140-1145.
[2] DAS K C,GUTMAN I,FURTULA B.On spectral radius and energy of extended adjacency matrix of graphs [J].Applied Mathematics and Computation,2017,296:116-123.
[3] GHORBANI M,LI X L,ZANGI S,et al.On the eigenvalue and energy of extended adjacency matrix [J].Applied Mathematics and Computation,2021,397:125939.
[4] LIU C,PAN Y G,DAI L,et al.Some eigenvalue properties and new bounds for the energy of extended adjacency matrix of graphs [J].MATCH Commun Math Comput Chem,2020,84(2):349-362.
[5] WANG Z,MAO Y P,FURTULA B,et al.Bounds for the spectral radius and energy of extended adjacency matrix of graphs [J].Linear and Multilinear Algebra,2021,69(10):1813-1824.
[6] RAD N J,JAHANBANI A,GUTMAN I.Zagreb energy and Zagreb estrada index of graphs [J].MATCH-Communications in Mathematical and in Computer Chemistry,2018,79(2):371-386.
[7] GAO Y B,SHAO Y L.The minimum ABC energy of trees [J].Linear Algebra and Its Applications,2019,577:186-203.
[8] GHORBANI M,LI X L,HAKIMI-NEZHAAD M,et al.Bounds on the ABC spectral radius and ABC energy of graphs [J].Linear Algebra and Its Applications,2020,598:145-164.
[9] GUO X,GAO Y B.Arithmetic-geometric spectral radius and energy of graphs [J].MACTH Commun Math Comput Chem,2020,83(3):651-660.
[10] XU B G,LI S C,RONG Y,et al.On the spectral radius and energy of the weighted adjacency matrix of a graph [J].Applied Mathematics & Computation,2019,340:156-163.
[11] AOUCHICHE M,HANSEN P.A survey of Nordhaus-Gaddum type relations [J].Discrete Applied Mathematics,2013,161(4/5):466-546.
[12] ZHENG L,TIAN G X,CUI S Y.On spectral radius and energy of arithmetic-geometric matrix of graphs [J].MACTH Commun Math Comput Chem,2020,83(3):635-650.
[13] JAHANBANI A.New Bounds for the harmonic energy and harmonic estrada index of graphs [J].Computer Science Journal of Moldova,2018,78(3):270-300.
[14] GUTMAN I,RODRÍGUEZ J M,SIGARRETA J M.Linear and non-linear inequalities on the inverse sum indeg index [J].Discrete Applied Mathematics,2019,258:123-134.