1. Some results on the Aα-eigenvalues of a graph.
- Author
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Chen, Hongzhang, Li, Jianxi, and Chee Shiu, Wai
- Subjects
- *
LINEAR orderings , *REGULAR graphs , *EIGENVALUES , *SUBDIVISION surfaces (Geometry) - Abstract
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]
- Published
- 2023
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