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

Abstract

The \(A_\alpha \)-spectral radius of a graph G is the largest eigenvalue of \(A_\alpha (G):=\alpha D(G)+(1-\alpha ) A(G)\) for any real number \(\alpha \in [0,1]\), 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 \(A_\alpha \)-spectral radius over \({\mathscr {B}}(n, k)\), the class of clique trees of order n with a zero forcing number k, where \(0 \le \alpha <1\), \(\left\lfloor \frac{n}{2}\right\rfloor +1 \le k \le n-1\) and each block is a clique of size at least 3. Moreover, an estimation on the \(A_\alpha \)-spectral radius of the extremal graph is also included. Our result covers a recent result of Das (2023).