Citation: Cheng, Y.-J.; Kao, L.; Weng, C.-w. Corrigendum of a Theorem on New Upper Bounds for the α-Indices. Comment on Lenes et al. New Bounds for the α-Indices of Graphs. Mathematics 2020, 8, 1668. Mathematics 2022, 10, 2619. https:// doi.org/10.3390/math10152619 Received: 7 June 2022 Accepted: 25 July 2022 Published: 27 July 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Comment Corrigendum of a Theorem on New Upper Bounds for the α-Indices. Comment on Lenes et al. New Bounds for the α-Indices of Graphs. Mathematics 2020, 8, 1668 Yen-Jen Cheng *, Louis Kao and Chih-wen Weng Department of Applied Mathematics, National Yang Ming Chiao Tung University, 1001 University Road, Hsinchu 30010, Taiwan; chihpengkao.am03g@g2.nctu.edu.tw (L.K.); weng@math.nctu.edu.tw (C.-w.W.) * Correspondence: yjc7755@nycu.edu.tw Abstract: We give a family of counterexamples of a theorem on a new upper bound for the α-indices of graphs in the paper that is mentioned in the title. We also provide a new upper bound for corrigendum. Keywords: nonnegative matrix; graph; index; α-index MSC: 05C50; 15A42 The Statement, Counterexamples, and Corrigendum Let C be an n × n real symmetric matrix. The index of C, denoted by ρ(C), is the largest eigenvalue of C. Let G =(V, E) be a connected graph of order n = |V| and size m = | E| with adjacency matrix A(G) and diagonal matrix D(G) of degree sequence. Nikiforov [1] proposed the following matrix: A α (G)= αD(G)+(1 − α) A(G), where 0 ≤ α ≤ 1. The α-index of G, denoted by ρ α (G), is the index of A α (G). E. Lenes, E. Mallea-Zepeda, and J. Rodríguez [2] (Theorem 4) gave the following upper bound for ρ α (G). ρ α (G) ≤ δ − 1 + α +  (δ + 1 − α) 2 + 4(2m − nδ)(1 − α) 2 , (1) where δ is the minimum degree of G. The upper bound of ρ α (G) in (1) is not true by the following family of counterexamples. Example 1. It was shown in [1] that the α-index of the star graph K 1,n−1 is ρ α (K 1,n−1 )= 1 2  αn +  α 2 n 2 + 4(n − 1)(1 − 2α)  . Suppose α  = 0. Then lim n→∞ ρ α (K 1,n−1 ) αn = 1. (2) Applying δ = 1 for K 1,n−1 with n ≥ 2 in (1), we find ρ α (K 1,n−1 ) ≤ 1 2  α +  (2 − α) 2 + 4(n − 2)(1 − α)  ∼ n 1 2 . Hence lim n→∞ ρ α (K 1,n−1 ) αn = 0, Mathematics 2022, 10, 2619. https://doi.org/10.3390/math10152619 https://www.mdpi.com/journal/mathematics