On the largest real root of independence polynomials of graphs, an ordering on graphs, and starlike trees

Let $G$ be a simple graph of order $n$. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of $G$ is the polynomial $I(G,x)={\sum }_{k=0}^{n}s(G,k)x^k$, where $s(G,k)$ is the number of independent sets of $G$ of size $k$ and $s(G,0)=1$. Clearly all real roots of $I(G,x)$ are negative. Let $\xi (G)$ be the largest real root of $I(G,x)$. Let $H$ be a simple graph. By $G⪰H$ we mean that $I(H,x)\geq I(G,x)$forevery$x$intheinterval$[\xi(G),0]$\mathrm{.}Wenotethat$G\succeq H$impliesthat$\xi(G)\geq \xi(H)$\mathrm{.}Alsowelet$G\succ H$ifandonlyif$G\succeq H$and$I(G,x)\neq I(H,x)$\mathrm{.}Weprovethatforeverytree$T$oforder$n$,$S_n\succeq T\succeq P_n$,where$S_n$and$P_n$ are the star and the path of order n, respectively. By $T=T(n_1,\dots ,n_k)$ we mean a tree $T$ which has a vertex $v$ of degree $k$ such that $T\setminus v=P_{n_1-1}+\cdots +P_{n_k-1}$, that is $T\setminus v$ is the disjoint union of the paths $P_{n_1-1},\dots ,P_{n_k-1}$. Let $X=(x_1,\dots ,x_k)$ and $Y=(y_1,\dots ,y_k)$, where $x_1\ge \cdots \ge x_k$ and $y_1\geq\cdots\geq y_k$arereal\mathrm{.}By$X\succ Y$,wemean$x_1=y_1,\ldots,x_{t-1}=y_{t-1}$and$x_t>y_t$forsome$t\in{1,\ldots,k}$\mathrm{.}Welet$X\succ_{d}Y$,if$X\neq Y$ and for every $j$, $1\le j\le k$, ${\sum }_{i=1}^j{x}_i\ge {\sum }_{i=1}^j{y}_i$. Among all trees with fixed number of vertices, we show that if $(m_1,\dots ,m_k){\succ }_d(n_1,\dots ,n_k)$, then $T(n_1,\ldots,n_k)\succ T(m_1,\ldots,m_k)$\mathrm{.}Weconjecturethat$T(n_1,\ldots,n_k)\succ T(m_1,\ldots,m_k)$ifandonlyif$(m_1,\ldots,m_k)\succ (n_1,\ldots,n_k)$, where ${\sum }_{i=1}^k{n}_i={\sum }_{i=1}^k{m}_i$.