Geodesic
Graphs Whose Aα-Spectral Radius Does Not Exceed 2
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 677-690

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Let A(G) and D(G) be the adjacency matrix and the degree matrix of a graph G, respectively. For any real α ∈ [0, 1], we consider Aα(G) = αD(G) + (1 − α)A(G) as a graph matrix, whose largest eigenvalue is called the Aα-spectral radius of G. We first show that the smallest limit point for the Aα-spectral radius of graphs is 2, and then we characterize the connected graphs whose Aα-spectral radius is at most 2. Finally, we show that all such graphs, with four exceptions, are determined by their Aα-spectra.
Keywords: Aα -matrix, Smith graphs, limit point, spectral radius, index 
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Wang, Jian Feng; Wang, Jing; Liu, Xiaogang; Belardo, Francesco. Graphs Whose Aα-Spectral Radius Does Not Exceed 2. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 677-690. http://geodesic.mathdoc.fr/item/DMGT_2020_40_2_a19/