Bounds for the signless Laplacian energy of digraphs
type: Item
instance of: scholarly article
title: Bounds for the signless Laplacian energy of digraphs
zbMATH Open document ID: 1385.05053
author: Weige Xi
author: Ligong Wang
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2018-02-27T00:00:00Z
review text: The paper under review gives upper and lower bounds for the signless Laplacian energy of finite directed graphs without loops and multiple arcs but perhaps with a pair of oppositely directed arcs joining the same pair of vertices. The signless Laplacian is defined to be \(Q=D+A,\) where \(D\) is the diagonal matrix with outdegrees of vertices along the diagonal and \(A\) is the adjacency matrix. The energy of the signless Laplacian is defined to be the sum of \(\left| q_{i}-\frac{a}{n}\right| ,\) where \(q_{i}\) is the \(i\)th eigenvalue of the signless Laplacian, while \(a\) is the number of arcs and \(n\) the number of vertices. For example, Theorem 2.4 says that, if \(d_{i}^{+}\) denotes the outdegree of the \(i\)th vertex, the energy of the signless Laplacian is less than or equal to \(\;\sqrt{M_{1}n},\) where \(M_{1}=a+\sum \limits_{i=1}^{n}\sum\limits_{i=1}^{n} \left( d_{i}^{+}-\frac{a}{n}\right) ^{2}.\) Theorem 2.7 says that if \(c_{2}\) is the number of directed closed walks of length \(2,\) the energy of the signless Laplacian is greater than or equal to \(\sqrt{2M},\) where \(M=c_{2}+M_{1}-a.\)
reviewed by: Audrey A. Terras
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C20
zbMATH DE Number: 6843790
zbMATH Keywords: digraph
zbMATH Keywords: signless Laplacian energy
zbMATH Keywords: energy
zbMATH Keywords: eigenvalues
MaRDI profile type: Publication
full work available at URL: s13226-017-0233-8
OpenAlex ID: W2752973603
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DOI: 10.1007/S13226-017-0233-8
description: scientific article; zbMATH DE number 6843790