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Title: | On the difference between the spectral radius and the maximum degree of graphs∗ |
Authors: | Oboudi, M.R. |
Keywords: | tree eigenvalues of graphs spectral radius of graphs maximum degree |
Issue Date: | 2017 |
Publisher: | ДЗ "ЛНУ імені Тараса Шевченка" |
Series/Report no.: | математичні науки; |
Abstract: | Let G be a graph with the eigenvalues λ1(G) >· · · > λn(G). The largest eigenvalue of G, λ1(G), is called the spectral radius of G. Let β(G) = ∆(G) − λ1(G), where ∆(G) is the maximum degree of vertices of G. It is known that if G is a connected graph, then β(G) > 0 and the equality holds if and only if G is regular. In this paper we study the maximum value and the minimum value of β(G) among all non-regular connected graphs. In particular we show that for every tree T with n > 3 vertices, n − 1 − √ n − 1 > β(T) > 4 sin2 (π2n+2 ). Moreover, we prove that in the right side the equality holds if and only if T ∼= Pn and in the other side the equality holds if and only if T ∼= Sn, where Pn and Sn are the path and the star on n vertices, respectively. |
Description: | Oboudi M.R. On the difference between the spectral radius and the maximum degree of graphs / M.R.Oboudi // Algebra and Discrete Mathematics. - 2017. - Vol. 24. - Number 2. - Рp. 302-307 |
URI: | http://hdl.handle.net/123456789/4564 |
Appears in Collections: | Algebra and Discrete Mathematics. - № 2 (24). - 2017 |
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303-2040-2-PB.pdf | 287.36 kB | Adobe PDF | View/Open |
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