Let H be an abelian group written additively and k be a positive integer. Let G（H,k） denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sumcient condition for G（H, k1） ≈ G（H, k2）. We also discuss the problem when G（H1, k） is isomorphic to G（H2, k） for a given k. Moreover, we give an explicit formula of G（H, k） when H is a p-group and gcd（p, k） = 1.
We obtain all positive integer solutions（m1,m2,a,b） with a ＆gt; b,gcd（a,b） = 1 to the system of Diophantine equations km21- lat1bt2a2r= C1,km22- lat1bt2b2r= C2,with C1,C2 ∈ {-1,1,-2,2,-4,4},and k,l,t1,t2,r ∈ Z such that k ＆gt; 0,l ＆gt; 0,r ＆gt; 0,t1 ＆gt; 0,t2 0,gcd（k,l） = 1,and k is square-free.
Γ(x):=integral fromn=0 to ∞(e^(-t)t^(x-1)dt),x>0为gamma函数。设f(x):=logΓ(x)+logΓ(1-x),x∈Q(0,12]。证明如果存在有理数y0∈Q(0,12],使得f(y0)=logΓ(y0)+logΓ(1-y0)∈Q,则集合{eαπ|α∈珚Q}中恰好有一个代数数,即e-f(y0)π,且e-f(y0)π=sinπy0。
