论文标题
在树木的偏心矩阵上：惯性和光谱对称性
On the eccentricity matrices of trees: Inertia and spectral symmetry
论文作者
论文摘要
连接图$ g $的\ textit {偏心矩阵} $ \ MATHCAL {e}（g）$是从$ g $的距离矩阵中获得的，每行中的每一列和每列最大的非零条目，并在其余的零件中保留零。 $ \ mathcal {e}（g）$的特征值是$ g $的\ textIt {$ \ Mathcal {e} $ - eigenvalues}。在本文中，我们发现了树木怪异矩阵的惯性。有趣的是，任何超过$ 4 $的顶点的树具有奇数直径的两个正面和两个负$ \ Mathcal {e} $ - eigenvalues（不论树的结构如何）。直径均匀的树具有相同数量的正和负$ \ Mathcal {e} $ - eigenvalues，该values等于“直径为截然不克的”顶点的数量（请参见定义3.1）。此外，我们证明树的偏心矩阵的光谱相对于原点是对称的，并且仅当树的直径奇数时。作为一个应用程序，我们用三个不同的$ \ MATHCAL {E} $ - 特征来表征树。
The \textit{eccentricity matrix} $\mathcal{E}(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by keeping the largest non-zero entries in each row and each column, and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the \textit{$\mathcal{E}$-eigenvalues} of $G$. In this article, we find the inertia of the eccentricity matrices of trees. Interestingly, any tree on more than $4$ vertices with odd diameter has two positive and two negative $\mathcal{E}$-eigenvalues (irrespective of the structure of the tree). A tree with even diameter has the same number of positive and negative $\mathcal{E}$-eigenvalues, which is equal to the number of 'diametrically distinguished' vertices (see Definition 3.1). Besides we prove that the spectrum of the eccentricity matrix of a tree is symmetric with respect to the origin if and only if the tree has odd diameter. As an application, we characterize the trees with three distinct $\mathcal{E}$-eigenvalues.
