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Paper IPM / M / 14976  


Abstract:  
Let G be a graph with eigenvalues λ_{1}(G) ≥ … ≥ λ_{n}(G). The spectral radius of G is λ_{1}(G). Let β(G)=∆(G)−λ_{1}(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 nonregular connected graphs. In particular we show that for every tree T with n ≥ 3 vertices, n−1−√{n−1} ≥ β(T) ≥ 4sin^{2}([(π)/(2n+2)]). Moreover, we prove that in the right side the equality holds if and only if T ≅ P_{n} and in the other side the equality holds if and only if T ≅ S_{n}, where P_{n}, S_{n} are the path and the star on n vertices, respectively.
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