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| Paper IPM / M / 14976 |
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| Abstract: | |
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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 T ≅ Pn and in the other side the equality holds if and only if T ≅ Sn, where Pn, Sn are the path and the star on n vertices, respectively.
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