@article {IOPORT.06104577,
    author       = {Lan, Jingfen and Lu, Linyuan and Shi, Lingsheng},
    title        = {Graphs with diameter $n - e$ minimizing the spectral radius.},
    year         = {2012},
    journal      = {Linear Algebra and its Applications},
    volume       = {437},
    number       = {11},
    issn         = {0024-3795},
    pages        = {2823-2850},
    publisher    = {Elsevier Science Inc. (North-Holland), New York, NY},
    doi          = {10.1016/j.laa.2012.05.038},
    abstract     = {Summary: The spectral radius $\rho (G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix $A(G)$. For a fixed integer $e\geq 1$, let $G_{n,n-e}^{min}$ be a graph with minimal spectral radius among all connected graphs on $n$ vertices with diameter $n-e$. Let $P_{n_1,n_2,\dots,n_t,p}^{m_1,m_2,\dots,m_t}$ be a tree obtained from a path of $p$ vertices $(0\sim 1\sim 2\sim \dots \sim (p-1))$ by linking one pendant path $P_{n_{i}}$ at $m_{i}$ for each $i\in \{1,2,\ldots ,t\}$. For e=1,2,3,4,5, $G_{n,n-e}^{min}$ were determined in the literature. {\it S.M. Cioab\v a} et al. [Linear Algebra Appl. 432, No. 2--3, 722--737 (2010; Zbl 1209.05140)] conjectured for fixed $e\geq 6, G_{n,n-e}^{min}$ is in the family $\mathcal P_{n,e}=\{P_{2,1,\dots,1,2,n-e+1}^{2,m_2,\dots,m_{e-4},n-e-2}|2 <m_2<\dots<m_{e-4}<n-e-2\}$. For $e=6,7$, they conjectured $G_{n,n-6}^{min}=P_{2,1,2,n-5}^{2,\lceil\frac {D-1}{2}\rceil,D-2}$ and $G_{n,n-7}^{min}=P_{2,1,2,n-6}^{2,\lfloor\frac {D+2}{3}\rfloor,D-\lfloor\frac {D+2}{3}\rfloor D-2}$. In this paper, we settle their conjectures positively. Note that any tree in $P_{n,e}$ is uniquely determined by its internal path lengths. For any $e-4$ non-negative integers $k_{1},k_{2},\ldots ,k_{e-4}$, let $T_{(k_1,k_2,\dots,k_{n-4})}=P_{2,1,\dots,1,2,n-e+1}^{2,m_2,\dots,m_{e-4},n-e-2}$ with $k_{i}=m_{i+1}-m_{i}-1$, for $1\leq i\leq e-4$. (Here we assume $m_{1}=2$ and $m_{e-3}=n-e-2.$) Let $s=\frac {\sum_{i=1}^{e-4}k_{i}+2}{e-4}$. For any integer $e\geq 6$ and sufficiently large $n$, we proved that $G_{n,n-e}^{min}$ must be one of the trees $T_{(k_1,k_2,\dots,k_{n-4})}$ with the parameters satisfying $\lfloor s\rfloor -1\leq k_j\leq \lfloor s\rfloor \leq k_i\leq \lceil s\rceil+1$ for $j=1,e-4$ and $i=2,\ldots,e-5$. Moreover, $0\leq k_{i}-k_j\leq 2$ for $2\leq i\leq e-5, j=1, e-4$ and $|k_{i}-k_j|\leq 1$ for $2\leq i, j\leq e-5$. These results are best possible as shown by cases $e=6,7,8$, where $G_{n,n-e}^{min}$ are completely determined here. Moreover, if $n-6$ is divisible by $e-4$ and $n$ is sufficiently large, then $G_{n,e}^{min}=T_{(k-1,k,k,\dots,k,k,k-1)}$ where $k=\frac {n-6}{e-4}-2$.},
    identifier   = {06104577},
}