论文标题
在$ d $二维代数连接图上
On the $d$-dimensional algebraic connectivity of graphs
论文作者
论文摘要
The $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G=(V,E)$, introduced by Jordán and Tanigawa, is a quantitative measure of the $d$-dimensional rigidity of $G$ that is defined in terms of the eigenvalues of stiffness matrices (which are analogues of the graph Laplacian) associated to mappings of the vertex将$ v $设置为$ \ mathbb {r}^d $。 在这里，我们分析了完整图的$ d $维代数连接。特别是，我们表明，对于$ d \ geq 3 $，$ a_d（k_ {d+1}）= 1 $，对于$ n \ geq 2d $，\ [\ left \ left \ lceil \ frac {n} {2d} {2d} {2d} \ right \ right \ rceil-rceil-2d+1+1 \ 1 \ leq a_d \ leq leq \ frac {2n} {3（d-1）}+\ frac {1} {3}。 \]
The $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G=(V,E)$, introduced by Jordán and Tanigawa, is a quantitative measure of the $d$-dimensional rigidity of $G$ that is defined in terms of the eigenvalues of stiffness matrices (which are analogues of the graph Laplacian) associated to mappings of the vertex set $V$ into $\mathbb{R}^d$. Here, we analyze the $d$-dimensional algebraic connectivity of complete graphs. In particular, we show that, for $d\geq 3$, $a_d(K_{d+1})=1$, and for $n\geq 2d$, \[ \left\lceil\frac{n}{2d}\right\rceil-2d+1\leq a_d(K_n) \leq \frac{2n}{3(d-1)}+\frac{1}{3}. \]
