Merging the A- and Q-spectral theories

By Vladimir Nikiforov. 2016

304 citations recorded by the source corpus.

Let $G$ be a graph with adjacency matrix $A\left( G\right) $, and let $D\left( G\right) $ be the diagonal matrix of the degrees of $G.$ The signless Laplacian $Q\left( G\right) $ of $G$ is defined as $Q\left( G\right) :=A\left( G\right) +D\left( G\right) $. Cvetkovi\'{c} called the study of the adjacency matrix the $A$% \textit{-spectral theory}, and the study of the signless Laplacian--the $Q$\textit{-spectral theory}. During the years many similarities and differences between these two theories have been established. To track the gradual change of $A\left( G\right) $ into $Q\left( G\right) $ in this paper it is suggested to study the convex linear combinations $A_{\alpha }\left( G\right) $ of $A\left( G\right) $ and $D\left( G\right) $ defined by \[ A_{\alpha}\left( G\right) :=\alpha D\left( G\right) +\left( 1-\alpha\right) A\left( G\right) \text{, \ \ }0\leq\alpha\leq1. \] This study sheds new light on $A\left( G\right) $ and $Q\left( G\right) $, and yields some surprises, in particular, a novel spectral Tur\'{a}n theorem. A number of challenging open problems are discussed.

Subfields: Computational Theory and Mathematics, Electrical and Electronic Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9763551bfe5ece8c93ad01487713f829315a973d

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In Computational Theory and Mathematics

In Electrical and Electronic Engineering

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Citations in the corpus

On the α -index of graphs with pendent paths (2017, unknown): “In [9] the matrix Aα(G) has been defined for any real α ∈ [0, 1] as Aα(G) = αD(G) + (1 − α)A(G).”
On the $A_{\alpha}$-spectra of trees (2016, methodology): “The main tool in the proofs of Theorems 2 and 3 is Proposition 15 of [9], which reads as: Proposition 10 Let α ∈ [0, 1), and G be a graph with Aα (G) = Aα.”
On the A-characteristic polynomial of a graph (2017, background): “Up until now, a few properties on Aα(G) have been investigated, including bounds on the k-th largest (especially, the largest, the second largest and the smallest) eigenvalue of Aα(G) [12, 13], the positive semidefiniteness of Aα(G) [12, 14], etc.”
A note on the $A_{\alpha}$-spectral radius of graphs (2018, background): “The first lemma is due to Nikiforov and Rojo [Lemma 6, [9]], it turns out to be very useful in the paper.”