The strict spectral-radius minimization conjecture for block graphs

Let $\mathcal{B}(n,q)$ be the class of block graphs with parameters $n$ and $q$, let $P_b^q$ denote the corresponding $q$-clique path with $b$ blocks, and let $\prec$ be the graph comparison relation used in the paper. Strict spectral-radius minimization conjecture. If

$$G\in\mathcal{B}(n,q)\setminus\{P_b^q\},$$

then $G\prec P_b^q$. The conjecture is motivated by computational evidence that the clique path uniquely minimizes the relevant spectral-radius ordering among graphs in this class; the weaker consequence stated later in the paper is that $\rho(P_b^q)<\rho(G)$ for every other $G$ in the class.