For a graph \(G\), let \(A(G)\) denote its adjacency matrix and \(D(G)\) denote its diagonal degree matrix (i.e, \(D(G)\) is a diagonal matrix whose nonzero entries are the degrees of the vertices of \(G\)). For \(\alpha \in [0, 1]\), let \(A_{\alpha}(G) = \alpha D(G) + (1 - \alpha) A(G)\). Note \(A_0(G) = A(G)\), \(A_1(G) = D(G)\) and \(A_{\frac12}(G) = \frac12(D(G)+A(G)) = \frac12 Q(G)\), where \(Q(G)\) is the signless Laplacian matrix of \(G\). V. Nikiforov [Appl. Anal. Discrete Math. 11, No. 1, 81–107 (2017; Zbl 1499.05384)] proposed the study of the spectra of \(A_{\alpha}(G)\) for different values of \(\alpha\) as a common generalization of the study of the spectra of \(A(G)\) and \(Q(G)\).
In this paper, the authors study a number of different questions related to the spectra of the \(A_{\alpha}\)-matrices. In Section 2, they prove some results on the behavior of the \(A_{\alpha}\)-eigenvalues under the different graph operations of vertex deletion, edge deletion, vertex contraction, and edge subdivision.
In Section 3, the authors extend some previously known results of H. Liu and M. Lu [Linear Algebra Appl. 440, 83–89 (2014; Zbl 1285.05117)] relating the \(k\)-domination number and the eigenvalues of the signless Laplacian matrix \(Q\) to theorems relating the \(k\)-domination number and the eigenvalues of \(A_{\alpha}\).
In Section 4, the authors prove some results relating certain eigenvalues of the \(A_{\alpha}\) matrices to the independence number, chromatic number, and circumference of a graph.