arXiv 2008.059112020-08math.COcs.DM

关于Cayley图的二分性常数和扩张

On the Bipartiteness Constant and Expansion of Cayley Graphs

Nina Moorman, Peter Ralli, Prasad Tetali

备注：: 14 pages, 2 figures

摘要：

设$G$是一个有限的无向$d$-正则图，$A(G)$是它的归一化邻接矩阵，其特征值为$1 = λ_1(A)\geq \dots \ge λ_n \ge -1$. $λ_n = -1$当且仅当$G$是二分的，这是一个经典的事实.我们的主要结果提供了一个定量的分离$λ_n$从$-1$的情况下，凯莱图，在其扩展。表示$h_{out}$的（外边界）顶点扩展的$G$，我们表明，如果$G$是一个非二部凯莱图（构造使用一个组和一个对称生成集的大小$d$），然后$λ_n \ge -1 + ch_{out}^2/d^2\,,$的$c$的绝对常数。我们展示的图形，这一结果是紧到一个因素取决于$d$。这改进了Biswas和Saha最近的一个结果，他们证明了$λ_n \ge -1 + h_{out}^4/(2^9d^8)\,.$。我们还注意到，这样的结果对于一般的非二部图是不成立的。

英文摘要：

Let $G$ be a finite, undirected $d$-regular graph and $A(G)$ its normalized adjacency matrix, with eigenvalues $1 = λ_1(A)\geq \dots \ge λ_n \ge -1$. It is a classical fact that $λ_n = -1$ if and only if $G$ is bipartite. Our main result provides a quantitative separation of $λ_n$ from $-1$ in the case of Cayley graphs, in terms of their expansion. Denoting $h_{out}$ by the (outer boundary) vertex expansion of $G$, we show that if $G$ is a non-bipartite Cayley graph (constructed using a group and a symmetric generating set of size $d$) then $λ_n \ge -1 + ch_{out}^2/d^2\,,$ for $c$ an absolute constant. We exhibit graphs for which this result is tight up to a factor depending on $d$. This improves upon a recent result by Biswas and Saha who showed $λ_n \ge -1 + h_{out}^4/(2^9d^8)\,.$ We also note that such a result could not be true for general non-bipartite graphs.

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英文摘要：

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