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Let G be a simple graph of order n. For $ \alpha \in [0,1] $ α ∈ [ 0 , 1 ] , the $ A_{\alpha } $ A α -matrix of G is defined as $ A_{\alpha }(G)=\alpha D(G)+ (1-\alpha)A(G) $ A α (G) = α D (G) + (1 − α) A (G) , where $ A(G) $ A (G) and $ D(G) $ D (G) are the adjacency matrix and the diagonal degree matrix of G, respectively. The eigenvalues of the $ A_{\alpha } $ A α -matrix of G are called the $ A_{\alpha } $ A α -eigenvalues of G. In this paper, we first study the properties on $ A_{\alpha } $ A α -eigenvalues, i.e. how the $ A_{\alpha } $ A α -eigenvalues behave under some kinds of graph transformations including vertex deletion, vertex contraction, edge deletion and edge subdivision. Moreover, we also present the relationships between the $ A_{\alpha } $ A α -eigenvalues of G and its k-domination number, independence number, chromatic number and circumference, respectively. [ABSTRACT FROM AUTHOR]

The normalized Laplacian eigenvalues of a network play an important role in its structural and dynamical aspects associated with the network. In this paper, we consider how the normalized Laplacian spectral radius of a non-bipartite graph behaves by several graph operations. As an example of the application, the smallest normalized Laplacian spectral radius of non-bipartite unicyclic graphs with fixed order is determined. [ABSTRACT FROM PUBLISHER]