Bulletin of the
Korean Mathematical Society BKMS

In 2017, Nikiforov proposed the $A_{\alpha}$-matrix of a graph $G$. This novel matrix is defined as $$A_{\alpha}(G)=\alpha D(G)+(1- \alpha )A(G),~\alpha \in [0,1],$$ where $D(G)$ and $A(G)$ are the degree diagonal matrix and adjacency matrix of $G$, respectively. Recently, Zhai, Xue and Liu [39] considered the Brualdi-Hoffman-type problem for $Q$-spectra of graphs with given matching number. As a continuance of it, in this contribution we consider the Brualdi-Hoffman-type problem for $A_{\alpha}$-spectra of graphs with given matching number. We identify the graphs with given size and matching number having the largest $A_{\alpha}$-spectral radius for $\alpha \in [\frac{1}{2},1)$.

Keywords: $A_{\alpha}$-matrix, $A_{\alpha}$-spectral radius, size, matching number

MSC numbers: Primary 05C50

Supported by: This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 12171190, 11671164, 11971274).