#### Vol. 13, No. 2, 2020

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Connectedness of digraphs from quadratic polynomials

### Siji Chen and Sheng Chen

Vol. 13 (2020), No. 2, 357–360
##### Abstract

Suppose that $f\left(x\right)=x\left(x-k\right)$, where $k$ is an odd positive integer. First, an infinite digraph ${G}_{k}=\left(V,E\right)$ is defined, where the vertex set is $V=ℤ$ and the edge set is $E=\left\{\left(x,y\right)\mid x,y\in ℤ,f\left(x\right)=f\left(2y\right)\right\}$. Then the following results are proved: if $k=1$, then the digraph ${G}_{k}$ is weakly connected; if $p$ is a safe prime, i.e., both $p$ and $q=\left(p-1\right)∕2$ are primes, then the number ${w}_{p}$ of weakly connected components of the digraph ${G}_{p}$ is 2. Finally, a conjecture that there are infinitely many primes $p$ such that ${w}_{p}=2$ is presented.

##### Keywords
connectivity, digraph, prime
##### Mathematical Subject Classification 2010
Primary: 05C25
Secondary: 05C40, 05C20