论文标题
某些集团树的偏心矩阵的惯性和光谱对称性
Inertia and spectral symmetry of eccentricity matrices of some clique trees
论文作者
论文摘要
连接图$ g $的偏心矩阵$ \ mathcal e（g）$是从$ g $的距离矩阵中获得的，每行和每一列中最大的非零入口，并用零替换剩余的零件。在本文中，我们考虑了集团树的集合$ \ MATHCAL C \ MATHCAL T $，其块最多具有两个cut-vertices \ textColor {blue} {of Clique Tree}。在证明了$ \ Mathcal c \ Mathcal t $中的集团树的偏心矩阵并找到其惯性指数的偏心矩阵之后，我们表明，$ \ Mathcal c \ Mathcal t $中的每个图，具有超过$ 4 $ dimmeter的$ 4 $ dimmeter和odd odd timmeter和两个负面$ \ Mathcal e $ -E-eigiegues。正面$ \ Mathcal E $ -EIGENVALUES和负$ \ Mathcal E $ -EIGENVALUES的数量也相等，即使对于$ \ Mathcal C \ Mathcal c \ Mathcal t $，直径均匀;共享的基数还计算\ textColor {blue} {`damementally区分'}顶点。最后，我们证明了一个集团树的偏心矩阵的频谱$ g $ in $ \ MATHCAL C \ MATHCAL T $相对于原点是对称的，并且仅当$ G $具有奇数直径且正好是两个相邻的中央顶点。
The eccentricity matrix $\mathcal E(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by leaving unchanged the largest nonzero entries in each row and each column, and replacing the remaining ones with zeros. In this paper, we consider the set $\mathcal C \mathcal T$ of clique trees whose blocks have at most two cut-vertices \textcolor{blue}{of the clique tree}. After proving the irreducibility of the eccentricity matrix of a clique tree in $\mathcal C \mathcal T$ and finding its inertia indices, we show that every graph in $\mathcal C \mathcal T$ with more than $4$ vertices and odd diameter has two positive and two negative $\mathcal E$-eigenvalues. Positive $\mathcal E$-eigenvalues and negative $\mathcal E$-eigenvalues turn out to be equal in number even for graphs in $\mathcal C \mathcal T$ with even diameter; that shared cardinality also counts the \textcolor{blue}{`diametrally distinguished'} vertices. Finally, we prove that the spectrum of the eccentricity matrix of a clique tree $G$ in $\mathcal C \mathcal T$ is symmetric with respect to the origin if and only if $G$ has an odd diameter and exactly two adjacent central vertices.
