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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
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.

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