ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.06 https://doi.org/10.26493/1855-3974.2697.43a (Also available at http://amc-journal.eu) On the A α -spectral radius of connected graphs * Abdollah Alhevaz , Maryam Baghipur Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box: 316-3619995161, Shahrood, Iran Hilal Ahmad Ganie Department of School Education, JK Govt. Kashmir, India Kinkar Chandra Das † Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea Received 14 September 2021, accepted 23 June 2022, published online 5 October 2022 Abstract For a simple graph G, the generalized adjacency matrix A α (G) is defined as A α (G)= αD(G) + (1 − α)A(G),α ∈ [0, 1], where A(G) is the adjacency matrix and D(G) is the diagonal matrix of the vertex degrees. It is clear that A 0 (G)= A(G) and 2A 1 2 (G)= Q(G) implying that the matrix A α (G) is a generalization of the adjacency matrix and the signless Laplacian matrix. In this paper, we obtain some new upper and lower bounds for the generalized adjacency spectral radius λ(A α (G)), in terms of vertex degrees, average vertex 2-degrees, the order, the size, etc. The extremal graphs attaining these bounds are characterized. We will show that our bounds are better than some of the already known bounds for some classes of graphs. We derive a general upper bound for λ(A α (G)), in terms of vertex degrees and positive real numbers b i . As application, we obtain some new upper bounds for λ(A α (G)). Further, we obtain some relations between clique number ω(G), independence number γ (G) and the generalized adjacency eigenvalues of a graph G. Keywords: Adjacency matrix, signless Laplacian matrix, generalized adjacency matrix, spectral ra- dius, degree sequence, clique number, independence number. Math. Subj. Class. (2020): Primary: 05C50, 05C12; Secondary: 15A18. * The authors would like to thank the handling editor and two anonymous referees for their detailed constructive comments that helped improve the quality of the paper. † Corresponding author. Partially supported by the National Research Foundation of the Korean government with grant No. 2021R1F1A1050646. E-mail addresses: a.alhevaz@shahroodut.ac.ir (Abdollah Alhevaz), maryamb8989@gmail.com (Maryam Baghipur), hilahmad1119kt@gmail.com (Hilal Ahmad Ganie), kinkardas2003@gmail.com (Kinkar Chandra Das) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/