Coherently manipulating flying qubits in a quantum wire with a magnetic impurity

e study the effect of a magnetic impurity with spin-half on a single propagating electron in a one-dimensional model system via the tight-binding approach. Due to the spin-dependent interaction, the scattering channel for the flying qubit is split, and its transmission spectrum is obtained. It is found that, the spin orientation of the impurity plays the role as a spin state filter for a flying qubit.Comment: 6 pages, 5 figure

Graphs with Diameter n−en-e Minimizing the Spectral Radius

The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix $A(G)$. For a fixed integer $e\ge 1$, let $G^{min}_{n,n-e}$ be a graph with minimal spectral radius among all connected graphs on $n$ vertices with diameter $n-e$. Let $P_{n_1,n_2,...,n_t,p}^{m_1,m_2,...,m_t}$ be a tree obtained from a path of $p$ vertices ($0 \sim 1 \sim 2 \sim ... \sim (p-1)$) by linking one pendant path $P_{n_i}$ at $m_i$ for each $i\in\{1,2,...,t\}$. For $e=1,2,3,4,5$, $G^{min}_{n,n-e}$ were determined in the literature. Cioab\v{a}-van Dam-Koolen-Lee \cite{CDK} conjectured for fixed $e\geq 6$, $G^{min}_{n,n-e}$ is in the family ${\cal P}_{n,e}=\{P_{2,1,...1,2,n-e+1}^{2,m_2,...,m_{e-4},n-e-2}\mid 2<m_2<...<m_{e-4}<n-e-2\}$. For $e=6,7$, they conjectured $G^{min}_{n,n-6}=P^{2,\lceil\frac{D-1}{2}\rceil,D-2}_{2,1,2,n-5}$ and $G^{min}_{n,n-7}=P^{2,\lfloor\frac{D+2}{3}\rfloor,D- \lfloor\frac{D+2}{3}\rfloor, D-2}_{2,1,1,2,n-6}$. In this paper, we settle their three conjectures positively. We also determine $G^{min}_{n,n-8}$ in this paper