计算图的拉普拉斯能量、类拉普拉斯能量不变量和Kirchhoff指数

IF 0.3 Q4 COMPUTER SCIENCE, THEORY & METHODS Acta Universitatis Sapientiae Informatica Pub Date : 2022-12-01 DOI:10.2478/ausi-2022-0011
S. Bhatnagar, Merajuddin, S. Pirzada
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引用次数: 0

摘要

设G是一个n阶、大小为m的简单连通图,矩阵L(G)= D(G)−a (G)称为图G的拉普拉斯矩阵,其中D(G)和a (G)分别是度对角矩阵和邻接矩阵。设顶点度序列d1≥d2≥····dn,设μ1≥μ2≥···≥μn - 1 >μn = 0为G的拉普拉斯矩阵的特征值。图的不变量、拉普拉斯能量(LE)、类拉普拉斯能量不变量(LEL)和Kirchhoff指数(Kf)根据图G的拉普拉斯特征值定义为:LE=∑i=1n| μi-2mn | LE= \sum\nolimits _i =1{ ^n }{\left | {{\mu _i }-{{2m }\over n }}\right |;}LEL=∑i=1n-1μi LEL=\sum\nolimits _i =1{ ^}n -1{}{\sqrt{{\mu _i}}和Kf=n∑i=1n-11μi Kf=n }\sum\nolimits _i =1{ ^}n -{1}{{\over{{\mu _i}}}}。本文给出了类拉普拉斯能不变量LEL的一个新的界,建立了类拉普拉斯能不变量LEL与Kirchhoff指数Kf之间的关系。进一步,我们得到了拉普拉斯能量LE与基尔霍夫指数Kf之间的关系。
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Computing Laplacian energy, Laplacian-energy-like invariant and Kirchhoff index of graphs
Abstract Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian matrix of the graph G,where D(G) and A(G) are the degree diagonal matrix and the adjacency matrix, respectively. Let the vertex degree sequence be d1 ≥ d2 ≥··· ≥ dn and let μ1 ≥ μ2 ≥··· ≥ μn−1 >μn = 0 be the eigenvalues of the Laplacian matrix of G. The graph invariants, Laplacian energy (LE), the Laplacian-energy-like invariant (LEL) and the Kirchhoff index (Kf), are defined in terms of the Laplacian eigenvalues of graph G, as LE=∑i=1n| μi-2mn | LE = \sum\nolimits_{i = 1}^n {\left| {{\mu _i} - {{2m} \over n}} \right|} , LEL=∑i=1n-1μi LEL = \sum\nolimits_{i = 1}^{n - 1} {\sqrt {{\mu _i}} } and Kf=n∑i=1n-11μi Kf = n\sum\nolimits_{i = 1}^{n - 1} {{1 \over {{\mu _i}}}} respectively. In this paper, we obtain a new bound for the Laplacian-energy-like invariant LEL and establish the relations between Laplacian-energy-like invariant LEL and the Kirchhoff index Kf.Further,weobtain the relations between the Laplacian energy LE and Kirchhoff index Kf.
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Acta Universitatis Sapientiae Informatica
Acta Universitatis Sapientiae Informatica COMPUTER SCIENCE, THEORY & METHODS-
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