论文标题
在$ \ mathbb {z} _n $上的Paley型图上
On a Paley-type graph on $\mathbb{Z}_n$
论文作者
论文摘要
令$ q $为主要功率，以便$ q \ equiv 1 \ pmod {4} $。订单$ Q $的PAYY图是带有顶点集的图形，作为有限字段$ \ Mathbb {f} _Q $，而定义为，$ ab $的边缘是边缘，并且仅当$ a-b $是$ \ mathbb {f} _q $的$ a-b $是一个非零的平方。我们尝试构建类似的订单$ n $的图，其中$ n \ in \ mathbb {n} $。对于合适的$ n $，我们构造了图表，其中顶点集是有限的通勤戒指$ \ mathbb {z} _n $，而将某些单位$ x $ x $ $ x $ x $ x $ x $ x $ x $ x $ x $ x $ \ mathbb {z} z} _n $时，则为$ ab $是边缘。我们查看此图的某些属性。对于Primes $ p \ equiv 1 \ pmod {4} $，Evans，Pulham和Sheehan计算了Paley图中订单4的完整子图的数量。最近，道齐（Dawsey）和麦卡锡（McCarthy）在订单$ q $的广义Paley图中找到了订单4的完整子图的数量。在本文中，对于Primes $ p \ equiv 1 \ pmod {4} $和任何正整数$α$，我们在按$ \ mathbb {z} _ {p^α} $定义的图表中找到了订单3和4的完整子图的数量。
Let $q$ be a prime power such that $q\equiv 1\pmod{4}$. The Paley graph of order $q$ is the graph with vertex set as the finite field $\mathbb{F}_q$ and edges defined as, $ab$ is an edge if and only if $a-b$ is a non-zero square in $\mathbb{F}_q$. We attempt to construct a similar graph of order $n$, where $n\in\mathbb{N}$. For suitable $n$, we construct the graph where the vertex set is the finite commutative ring $\mathbb{Z}_n$ and edges defined as, $ab$ is an edge if and only if $a-b\equiv x^2\pmod{n}$ for some unit $x$ of $\mathbb{Z}_n$. We look at some properties of this graph. For primes $p\equiv 1\pmod{4}$, Evans, Pulham and Sheehan computed the number of complete subgraphs of order 4 in the Paley graph. Very recently, Dawsey and McCarthy find the number of complete subgraphs of order 4 in the generalized Paley graph of order $q$. In this article, for primes $p\equiv 1\pmod{4}$ and any positive integer $α$, we find the number of complete subgraphs of order 3 and 4 in our graph defined over $\mathbb{Z}_{p^α}$.
