2 problems

Let $n_1\geq\cdots\geq n_k\geq 2$ and $m_1\geq\cdots\geq m_k\geq 2$. Let $T_1=T(n_1,\ldots,n_k)$ and $T_2=T(m_1,\ldots,m_k)$ be starlike trees of order $n$ and maximum degree $k$.…

Let $\succeq$ be the order on graphs defined by comparing their independence polynomials, and let $\mathcal{T}_n$ be the set of trees with $n$ vertices. Total-order conjecture. For…