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Paper   IPM / M / 14976
 School of Mathematics Title: On the difference between the spectral radius and the maximum degree of graphs Author(s): Mohammad Reza Oboudi Status: Published Journal: Algebra Discrete Math. Vol.: 24 Year: 2017 Pages: 302-307 Supported by: IPM
Abstract:
Let G be a graph with eigenvalues λ1(G) ≥ … ≥ λn(G). The spectral radius of G is λ1(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) ≥ 4sin2([(π)/(2n+2)]). Moreover, we prove that in the right side the equality holds if and only if TPn and in the other side the equality holds if and only if TSn, where Pn, Sn are the path and the star on n vertices, respectively.        