论文标题
直径图的补充的距离光谱大于三
The distance spectrum of the complements of graphs of diameter greater than three
论文作者
论文摘要
假设$ g $是一个连接的简单图，带有顶点集$ v（g）= \ {v_1，v_2，\ cdots，v_n \} $。令$ d（v_i，v_j）$为$ v_i $和$ v_j $之间的距离。然后$ g $的距离矩阵是$ d（g）=（d_ {ij}）_ {n \ times n} $，其中$ d_ {ij} = d（v_i，v_i，v_j）$。由于$ d（g）$是一种非负的真实对称矩阵，因此可以安排其特征值$λ_1（g）\geλ_2（g）\ ge \ ge \ cdots \geλ_n（g）$分别$ g $。图$ g $的{\ it直径}是所有对顶点之间的最远距离。在本文中，我们确定了距离光谱半径的独特图，在所有直径图的所有互补符中都达到了最大和最小值。此外，我们还表征了独特的图，其最小距离特征值分别在所有大于三的直径图的互补符中达到最大和最小值。
Suppose that $G$ is a connected simple graph with the vertex set $V( G ) = \{ v_1,v_2,\cdots ,v_n \} $. Let $d( v_i,v_j ) $ be the distance between $v_i$ and $v_j$. Then the distance matrix of $G$ is $D( G ) =( d_{ij} )_{n\times n}$, where $d_{ij}=d( v_i,v_j ) $. Since $D( G )$ is a non-negative real symmetric matrix, its eigenvalues can be arranged $λ_1(G)\ge λ_2(G)\ge \cdots \ge λ_n(G)$, where eigenvalues $λ_1(G)$ and $λ_n(G)$ are called the distance spectral radius and the least distance eigenvalue of $G$, respectively. The {\it diameter} of graph $G$ is the farthest distance between all pairs of vertices. In this paper, we determine the unique graph whose distance spectral radius attains maximum and minimum among all complements of graphs of diameter greater than three, respectively. Furthermore, we also characterize the unique graph whose least distance eigenvalue attains maximum and minimum among all complements of graphs of diameter greater than three, respectively.
