论文标题
在树木的偏心矩阵的最大和最小特征值下
On the largest and least eigenvalues of eccentricity matrix of trees
论文作者
论文摘要
图$ g $的偏心矩阵$ \ varepsilon（g）$是从$ g $的距离矩阵中构建的，仅保留每行和每一列的最大距离。该矩阵可以解释为与从距离矩阵获得的邻接矩阵的对立面，仅保留每行和每一列等于1的距离。图$ g $的$ \ varepsilon $ - eigenvalues是其偏心矩阵$ \ varepsilon（g）$的$。 Wang等人\ cite {e}提出了确定带有给定顺序的最大$ \ varepsilon $ -spectral radius的问题。在本文中，我们考虑了上述$ n $ vertex树的问题。获得了$ n $ vertex树的最大$ \ varepsilon $ - 光谱半径，并获得了固定奇数直径的，并且还确定了相应的极端树。 $ [ - 2 \ sqrt {2}，0）$的至少$ \ varepsilon $ -eigenvalues的树已知。最后，我们在$ [-2- \ sqrt {13}，-2 \ sqrt {2}）$中确定至少$ \ varepsilon $ -eigenValues的树。
The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is constructed from the distance matrix of $G$ by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency matrix obtained from the distance matrix by keeping only the distances equal to 1 for each row and each column. The $\varepsilon$-eigenvalues of a graph $G$ are those of its eccentricity matrix $\varepsilon(G)$. Wang et al \cite{e} proposed the problem of determining the maximum $\varepsilon$-spectral radius of trees with given order. In this paper, we consider the above problem of $n$-vertex trees with given diameter. The maximum $\varepsilon$-spectral radius of $n$-vertex trees with fixed odd diameter is obtained, and the corresponding extremal trees are also determined. The trees with least $\varepsilon$-eigenvalues in $[-2\sqrt{2},0)$ have been known. Finally, we determine the trees with least $\varepsilon$-eigenvalues in $[-2-\sqrt{13},-2\sqrt{2})$.
