论文标题
在超级图和一些有限基团的超级图上
On the super graphs and reduced super graphs of some finite groups
论文作者
论文摘要
对于有限的组$ g $，让$ b $是$ g $上的等价（平等，共轭或订单）关系，让$ a $为a（功率，增强功率或通勤）图，带有顶点套装$ g $。 $ b $ super $ a $ graph是一个简单的图形，带有顶点套装$ g $，如果它们在相同的$ b $ - 等价类中，或者它们的$ b $等值类中的元素在原始$ a $ a $ agraph中相邻，则两个顶点相邻。通过删除从$ b $ super $ a $ a $图的主导顶点（与所有其他顶点相邻）获得的图表称为降低的$ b $ super $ a $ a $ agraph。在本文中，对于$ b $ super $ a $图的某些对，我们表征了一对图相等的有限组。我们还表征了$ g $的订单超级通勤图$δ^o（g）$的主要顶点，并证明对于$ n \ geq 4 $，身份元素是$δ^o（s_n）$和$δ^o（a_n）$的唯一主要顶点。我们表征了$ n $的值，$ n $的值超级通勤图$Δ^o（s_n）^*$ of $ s_n $和还原订单超级通勤图$δ^o（a_n）^*$ a_n $的$ a_n）^*$。我们还证明，如果连接$δ^o（s_n）^*$（或$δ^o（a_n）^*$），则直径最多为$ 3 $，并且表明直径为$ 3 $，许多值的值为$n。$。
For a finite group $G$, let $B$ be an equivalence (equality, conjugacy or order) relation on $G$ and let $A$ be a (power, enhanced power or commuting) graph with vertex set $G$. The $B$ super $A$ graph is a simple graph with vertex set $G$ and two vertices are adjacent if either they are in the same $B$-equivalence class or there are elements in their $B$-equivalence classes that are adjacent in the original $A$ graph. The graph obtained by deleting the dominant vertices (adjacent to all other vertices) from a $B$ super $A$ graph is called the reduced $B$ super $A$ graph. In this article, for some pairs of $B$ super $A$ graphs, we characterize the finite groups for which a pair of graphs are equal. We also characterize the dominant vertices for the order super commuting graph $Δ^o(G)$ of $G$ and prove that for $n\geq 4$ the identity element is the only dominant vertex of $Δ^o(S_n)$ and $Δ^o(A_n)$. We characterize the values of $n$ for which the reduced order super commuting graph $Δ^o(S_n)^*$ of $S_n$ and the reduced order super commuting graph $Δ^o(A_n)^*$ of $A_n$ are connected. We also prove that if $Δ^o(S_n)^*$ (or $Δ^o(A_n)^*$) is connected then the diameter is at most $3$ and shown that the diameter is $3$ for many value of $n.$
