A Classification of Ramanujan Unitary Cayley Graphs
A Classification of Ramanujan Unitary Cayley Graphs
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拉马努金酉凯莱图的分类
DOI：
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发表时间：
2010
影响因子：
0.7
通讯作者：
Andrew Droll
中科院分区：
文献类型：
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作者：
Andrew Droll
n$个顶点的酉Cayley图$X_n$有顶点集${Bbb Z}/{nBbb Z}$，两个顶点$a$和$B$有边连通当且仅当它们相差一个模$n$的乘法单位，即${
m gcd}（a-b，n）= 1。一个k-正则图X是Ramanujan图当且仅当$lambda（X）leq 2sqrt{k-1}$其中$lambda（X）$是X$的邻接矩阵的特征值的第二大绝对值。我们得到了酉Cayley图X_n$是Ramanujan图的一个完全刻画。
The unitary Cayley graph on $n$ vertices, $X_n$, has vertex set ${Bbb Z}/{nBbb Z}$, and two vertices $a$ and $b$ are connected by an edge if and only if they differ by a multiplicative unit modulo $n$, i.e. ${
m gcd}(a-b,n) = 1$. A $k$-regular graph $X$ is Ramanujan if and only if $lambda(X) leq 2sqrt{k-1}$ where $lambda(X)$ is the second largest absolute value of the eigenvalues of the adjacency matrix of $X$. We obtain a complete characterization of the cases in which the unitary Cayley graph $X_n$ is a Ramanujan graph.