Nikiforov–Rojo conjectures on the $A_{b1}$-spectral radius of graphs with pendent paths

Let $G$ be a connected graph and let $u,v\tnobreak$ be vertices with $d(u),d(v)\geq 2$. Suppose that $u$ and $v$ are joined by a path

$$w_0(=v)w_1\cdots w_{s-1}w_s(=u),$$

where $d(w_i)=2$ for $1\leq i\leq s-1$. Let $G_{p,s,q}(u,v)$ be the graph obtained by attaching the paths $P_p$ to $u$ and $P_q$ to $v$, and let $\rho_{\alpha}(G)$ denote the spectral radius of $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$. Nikiforov–Rojo's conjectures. For $0\leq\alpha<1$ and $s=0,1$, if $p\geq q+2$, then

$$\rho_{\alpha}(G_{p,s,q}(u,v))<\rho_{\alpha}(G_{p-1,s,q+1}(u,v)).$$

The claim is settled by the theorem proved in this paper; it was also independently confirmed by Guo and Zhou.