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Atlas theory is an important part of the graph theory , which includes adjacent spectral theory , Laplace spectral theory , intended to Laplace spectral theory . In these spectra with Laplacian spectrum reflect the nature and structure of the Figure is the most convenient. S is a graph G is a set of ideas , if any two points in S are not adjacent , then S is an independent set . Independent set of maximum cardinality is called the independence number of a graph G , denoted by alpha ( G ) . | E ( G ) | = | V ( G ) │ connected graph G said bicyclic graphs . Hutchison order n , the number of independent alpha double circle picture shows ( ? ) ( N , alpha ) , ( ? ) 1 ( n , alpha ) ( ? ) ( N , alpha ) with two edge disjoint ring bicyclic graphs ( ? ) 2 ( n , a ) = ( ? ) ( n , alpha ) \\ ( ? ) 1 ( n alpha ) . In this paper, the Laplacian spectra of the structure and properties of a given number of independent . Firstly, the Laplacian spectral radius sector and Laplacian spectral radius of the given parameter pole figure depicts the results . Second , we characterize the n-order simple connected graph , independence number a ( G ) ∈ {1,2 , [ n / 2 ] , [ n / 2 ] 1 , n -3 , n - 2 , n - 1 } with the smallest Laplacian spectral radius of the pole figure . Finally , for any number of independent alpha were portrayed ( ? ) With the largest Laplacian spectral radius of the pole figure 1 ( n , alpha ) , ( ? ) 2 ( n , alpha ) , and further ( ? ) ( n , alpha ) Laplacian spectral radius of the sector and the pole figure .
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