Clique trees with a given zero forcing number maximizing the \(A_\alpha \)-spectral radius

The \(A_\alpha \) $A_{\alpha }$ -spectral radius of a graph G is the largest eigenvalue of \(A_\alpha (G):=\alpha D(G)+(1-\alpha ) A(G)\) $A_{\alpha }(G):=\alpha D(G)+(1-\alpha )A(G)$ for any real number \(\alpha \in [0,1]\) $\alpha \in [0,1]$ , where A(G) and D(G) are the adjacency matrix and the degree matrix of G, respectively. In this paper, we settle the problem of characterizing graphs which attain the maximum \(A_\alpha \) $A_{\alpha }$ -spectral radius over \({\mathscr {B}}(n, k)\) $\mathcal{B}(n,k)$ , the class of clique trees of order n with a zero forcing number k, where \(0 \le \alpha <1\) $0\le \alpha <1$ , \(\left\lfloor \frac{n}{2}\right\rfloor +1 \le k \le n-1\) $⌊\frac{n}{2}⌋+1\le k\le n-1$ and each block is a clique of size at least 3. Moreover, an estimation on the \(A_\alpha \) $A_{\alpha }$ -spectral radius of the extremal graph is also included. Our result covers a recent result of Das (2023).