The largest eigenvalue of unicyclic graphs
type: Item
instance of: scholarly article
title: The largest eigenvalue of unicyclic graphs
zbMATH Open document ID: 1116.05050
author: Shengbiao Hu
published in: Discrete Mathematics
publication date: 2007-02-13T00:00:00Z
review text: The author shows that the largest eigenvalue of the adjacency matrix of a unicyclic graph with the maximum vertex degree \(\Delta\) is bounded from above by \(2\sqrt{\Delta-1}\), while the largest eigenvalue of its Laplacian matrix is bounded by \(\Delta+2\sqrt{\Delta-1}\), with equality in the first case holding for all cycles, and in the second case for even cycles. These results improve previous results of \textit{C. D. Godsil} [Ann. Discrete Math. 20, 151-159 (1984; Zbl 0559.05040)] and of the reviewer [Linear Algebra Appl. 360, 35-42 (2003; Zbl 1028.05062)] on the largest eigenvalue of trees with fixed maximum degree.
reviewed by: Dragan Stevanović
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 05C38
zbMATH DE Number: 5124995
zbMATH Keywords: adjacency matrix
zbMATH Keywords: Laplacian matrix
MaRDI profile type: Publication
full work available at URL: j.disc.2006.06.023
OpenAlex ID: W2052353854
cites work: Q5422499
cites work: Q3347919
cites work: A relation between the matching number and Laplacian spectrum of a graph
cites work: Bounding the largest eigenvalue of trees in terms of the largest vertex degree
DOI: 10.1016/J.DISC.2006.06.023
description: scientific article; zbMATH DE number 5124995