Some extremal results on the Aα-spectral radius of the complement graphs of trees
Let A(G)and D(G)denote the adjacency matrix and the diagonal matrix of the degrees of a simple graph G,respectively.For α ∈[0,1),let Aα(G)=αD(G)+(1-α)A(G)be the Aα-matrix of the graph G,and the largest eigenvalue of Aα-matrix is called the Aα-spectral radius of G.The Aα-matrix of a graph G is a unified definition of the adjacency matrix and the signless Laplacian matrix of G.In this paper,the unique maximal and unique minimal extremal graph in the class of the complement graphs of trees with n vertices and maximum degree Δ is determined,respectively.Consequently,the unique minimal graph in the class of the complement graphs of trees with n vertices is also determined.The Scalar Theorem of spectral radius in the class of the complement graphs of trees with n vertices is proved.
Aα-matrixspectral radiustreescomplement of graphThe Scalar Theorem
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