Upper bounds on the spectral radius of book-free and/or \(K_{2,l}\)-free graphs
type: Item
instance of: scholarly article
title: Upper bounds on the spectral radius of book-free and/or \(K_{2,l}\)-free graphs
zbMATH Open document ID: 1109.05075
author: Zhipeng Song
author: Lingsheng Shi
published in: Linear Algebra and its Applications
publication date: 2007-01-09T00:00:00Z
review text: In this paper, the authors obtain upper bounds on the spectral radius of book-free and/or \(K_{2,l}\)-free graphs. Let \(G=(V,E)\) be a graph of order \(n\). The spectral radius \(\rho(G)\) of \(G\) is the largest eigenvalue of its adjacency matrix. For \(v \in V\), the degree of \(v\) is \(d(v)=| N(v)| \), where \(N(v)\) is the set of vertices adjacent to \(v\). The maximum degree of \(G\) is \(\Delta(G)=\max\{d(v) : v \in V \}\). A \(k\)-regular graph, that is a graph whose vertices have the same degree \(k\), is called strongly regular with parameters \((k,a,c)\) whenever each pair of adjacent vertices have \(a \geq 0\) common neighbors, and each pair of nonadjacent vertices have \(c \geq 1\) common neighbors. The main result of this paper establishs that if \(0\leq k \leq l \leq \Delta < n\) and \(G\) is a connected \(\{B_{k+1},K_{2,l+1}\}\)-free graph of order \(n\) with maximum degree \(\Delta\), then \[ \rho(G) \leq \left[k-l+\sqrt{(k-l)^2+4 \Delta +4l(n-1)} \right]/2, \] where \(B_{k+1}\) denotes a book with \(k+1\) pages consisting of \(k+1\) triangles sharing one edge. The equality, in the above expression, is satisfied if and only if \(G\) is a strongly regular graph with parameters \((\Delta, k, l)\).
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 5083621
zbMATH Keywords: girth
zbMATH Keywords: strongly regular graph
reviewed by: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: j.laa.2006.08.007
OpenAlex ID: W1976471160
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DOI: 10.1016/J.LAA.2006.08.007
description: scientific article; zbMATH DE number 5083621