The maximal b1-spectral-radius conjecture for bicyclic graphs

Let $\frac{1}{2}<\alpha<1$, and let $\mathscr{B}(n,d)$ denote the family of bicyclic graphs with order $n$ and diameter $d$. For a graph $G$, write its $\alpha$-spectral radius as the largest eigenvalue of its $\alpha$-matrix. Let $B_3^*(n,d)$ be the specified bicyclic graph in the paper.

Maximal $\alpha$-spectral-radius conjecture. If $G$ has the maximal $\alpha$-spectral radius among graphs in $\mathscr{B}(n,d)$, then

$$G\cong B_3^*(n,d).$$

This conjecture extends the paper's extremal results for the signless Laplacian spectral radius and is motivated by those theorems and numerical calculations. Its status is not resolved in the supplied text.