On the $A_{\alpha}$-spectra of trees

By Vladimir Nikiforov, Germain Pastén, Oscar Rojo, Ricardo L. Soto. 2016

68 citations recorded by the source corpus.

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\alpha\in\left[ 0,1\right],$ define the matrix $A_{\alpha}\left(G\right) $ as \[ A_{\alpha}\left(G\right) =\alpha D\left(G\right) +(1-\alpha)A\left(G\right) \] where $0\leq\alpha\leq1$. This paper gives several results about the $A_{\alpha}$-matrices of trees. In particular, it is shown that if $T_{\Delta}$ is a tree of maximal degree $\Delta,$ then the spectral radius of $A_{\alpha}(T_{\Delta})$ satisfies the tight inequality \[ \rho(A_{\alpha}(T_{\Delta}))<\alpha\Delta+2(1-\alpha)\sqrt{\Delta-1}. \] This bound extends previous bounds of Godsil, Lovasz, and Stevanovic. The proof is based on some new results about the $A_{\alpha}$-matrices of Bethe trees and generalized Bethe trees. In addition, several bounds on the spectral radius of $A_{\alpha}$ of general graphs are proved, implying tight bounds for paths and Bethe trees.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d693862c4447356e929b8d594b52c976c7d2cc99

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Citations in the corpus

On the A-characteristic polynomial of a graph (2017, background): “So, it was claimed in [12, 13] that the matrices Aα(G) can underpin a unified theory of A(G) and Q(G).”
On the α -index of graphs with pendent paths (2017, background): “If ω = 2, then G must be a path, as the path it the graph with smallest α-index among connected graphs of given order (see [10]).”