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Bounds for the (Laplacian) spectral radius of graphs with parameter α

Gui-Xian Tian;  Ting-Zhu Huang — 2012

Czechoslovak Mathematical Journal

Let G be a simple connected graph of order n with degree sequence ( d 1 , d 2 , ... , d n ) . Denote ( α t ) i = ∑ j : i ∼ j d j α , ( α m ) i = ( α t ) i / d i α and ( α N ) i = ∑ j : i ∼ j ( α t ) j , where α is a real number. Denote by λ 1 ( G ) and μ 1 ( G ) the spectral radius of the adjacency matrix and the Laplacian matrix of G , respectively. In this paper, we present some upper and lower bounds of λ 1 ( G ) and μ 1 ( G ) in terms of ( α t ) i , ( α m ) i and ( α N ) i . Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.

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