The Aα-Spectral Radii of Graphs with Given Connectivity

Abstract

The A α -matrix is A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) with α ∈ [ 0 , 1 ] , given by Nikiforov in 2017, where A ( G ) is adjacent matrix, and D ( G ) is its diagonal matrix of the degrees of a graph G. The maximal eigenvalue of A α ( G ) is said to be the A α -spectral radius of G. In this work, we determine the graphs with largest A α ( G ) -spectral radius with fixed vertex or edge connectivity. In addition, related extremal graphs are characterized and equations satisfying A α ( G ) -spectral radius are proposed.

On the sum of the powers of $ A_\alpha $ eigenvalues of graphs and $ A_\alpha $-energy like invariant

“…As A α (G) is a symmetric matrix, for α ∈ 1 2 , 1 , clearly A α (G) is positive semidefinite and so the A α eigenvalues of G can be taken as ρ 1 (G) ≥ ρ 2 (G) ≥ • • • ≥ ρ n (G). In this setup, the matrices A(G), Q(G) and D(G) were seen in a new light and very interesting results were deduced in [3,10,11,14,17].…”

The $A_{\alpha}$-spectral Radius of Bicyclic Graphs with Given Degree Sequences

“…In order to explore the extent to which the summands of A(G) and D(G) determines the properties of Q(G), Nikiforov [11] in 2017 proposed to study the convex combinations A α -matrix of A(G) and D(G), and claimed in [12] that the matrices A α (G) can underpin a unified theory of A(G) and Q(G). In recent years, the research of A α -matrix is an intriguing topic in spectral graph theory, the reader may be referred to [5][6][7][8][9][10][12][13][14][15][16] and the references therein.…”

“…The extremal graph with maximal A α -spectral radius with fixed order and cut vertices, and the extremal tree with the maximal A α -spectral radius with fixed order and matching number are characterized by Lin et al in [7]. The extremal graphs with largest A α -spectral radius with fixed vertex or edge connectivity are depicted by Wang in [15]. Most recently, the extremal graphs with maximum A α -spectral radius among all graphs with given size (resp.…”

A Complete Characterization of Bidegreed Split Graphs with Four Distinct α-Eigenvalues

“…In particular, if α = 0, then A α (G) is exactly the adjacency matrix of G, and A α (G) = 1 2 Q(G) if α = 1 2 . We encourage the interested reader to consult [4][5][6][7][8][9][10][11][12][13] and the references therein for more mathematical properties of A α (G).…”