A Sharp Upper Bound of the Spectral Radius of Graphs
TL;DR: The following sharp upper bound is obtained of the minimum degree of vertices of G, which is either a regular graph or a bidegreed graph in which each vertex is of degree either ? or n?1.
About: This article is published in Journal of Combinatorial Theory, Series B.The article was published on 2001-03-01 and is currently open access. It has received 157 citations till now. The article focuses on the topics: Bound graph & Graph power.
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TL;DR: In this paper, the spectral radius of the adjacency matrix and the Laplacian matrix of a simple undirected graph is analyzed in terms of the degrees and the 2-degrees of vertices.
161 citations
Cites background from "A Sharp Upper Bound of the Spectral..."
...Up to now, many bounds for ρ(G) and μ(G) were given (see, for instance, [1–12]), but most of them are upper bounds....
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TL;DR: In this article, the spectral radius of a Turan graph of order n was shown to be at most 2 m/n > 1 /( 2 m + 2 n ) unless G = T r ( n ).
138 citations
TL;DR: Some upper and lower bounds on the greatest eigenvalue and a lower bound on the smallest eigen value are presented.
104 citations
TL;DR: In this paper, the maximum spectral radius of graphs without paths of given length, and tight bounds on the spectral radius for graphs without given even cycles, were given. And they also raised a number of open problems.
82 citations
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TL;DR: In this paper, the maximum spectral radius of graphs without paths of given length and tight bounds on the spectral radius for graphs without given even cycles were given, and a number of natural open problems were raised.
Abstract: Let G be a graph with n vertices and mu(G) be the largest eigenvalue of the adjacency matrix of G. We study how large mu(G) can be when G does not contain cycles and paths of specified order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give tight bounds on the spectral radius of graphs without given even cycles. We also raise a number of natural open problems.
78 citations
Cites background from "A Sharp Upper Bound of the Spectral..."
...Fact 5 ([12]) If G is a graph with n vertices, m edges and δ (G) = δ, then µ (G) ≤ (δ − 1) /2 + √ 2m − δn + (δ + 1)2 /4, (9) Note that for connected graphs inequality (9) has been proved independently by Hong, Shu and Fang [10]....
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...2m − δn + (δ + 1)(2) /4, (9) Note that for connected graphs inequality (9) has been proved independently by Hong, Shu and Fang [10]....
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Book•
01 Jan 1976
TL;DR: In this paper, the authors present Graph Theory with Applications: Graph theory with applications, a collection of applications of graph theory in the field of Operational Research and Management. Journal of the Operational research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
Abstract: (1977). Graph Theory with Applications. Journal of the Operational Research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
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"A Sharp Upper Bound of the Spectral..." refers background in this paper
...The terminology not defined here can be found in [1, 3, 9, 11, 12]....
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TL;DR: In this article, the maximum spectral radius for (0, 1)-matrices with k2 and k2+1 1's, respectively, and for symmetric (1, 1) matrices with zero trace and e=k21's (graphs with e edges) was determined.
148 citations