论文标题
在图形的特征值$ d（5，q）$上
On the eigenvalues of the graphs $D(5, q)$
论文作者
论文摘要
令$ q = p^e $，其中$ p $是素数，$ e $是一个正整数。 Lazebnik和Ustimenko在1995年介绍了针对任何正整数$ k $和Prime Power $ Q $定义的图形$ D（K，Q）$。直到今天，图形$ d（k，q）$的连接组件为止，为给定的订单和给定订单和给定的Girth的图形大小提供了最低限制。此外，Ustimenko推测$ d（k，q）$的第二大特征值总是小于或等于$ 2 \ sqrt {q} $。如果是的，这意味着对于固定的$ q $和$ k $生长，$ d（k，q）$将定义一个几乎是Ramanujan的扩展器家族。在本文中，我们证明了猜想的最小开放式案例，表明对于所有奇数priper $ q $，第二大特征值$ d（5，q）$小于或等于$ 2 \ sqrt {q} $。
Let $q = p^e$, where $p$ is a prime and $e$ is a positive integer. The family of graphs $D(k, q)$, defined for any positive integer $k$ and prime power $q$, were introduced by Lazebnik and Ustimenko in 1995. To this day, the connected components of the graphs $D(k, q)$, provide the best known general lower bound for the size of a graph of given order and given girth. Furthermore, Ustimenko conjectured that the second largest eigenvalue of $D(k, q)$ is always less than or equal to $2\sqrt{q}$. If true, this would imply that for a fixed $q$ and $k$ growing, $D(k, q)$ would define a family of expanders that are nearly Ramanujan. In this paper we prove the smallest open case of the conjecture, showing that for all odd prime powers $q$, the second largest eigenvalue of $D(5, q)$ is less than or equal to $2\sqrt{q}$.
