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(x) = x(x k), where k is an odd positive integer. First, an infinite digraph Gk = (V,E) is defined, where the vertex set is V = and the edge set is E = {(x,y)x,y ,f(x) = f(2y)}. Then the following results are proved: if k = 1, then the digraph Gk is weakly connected; if p is a safe prime, i.e., both p and q = (p 1)2 are primes, then the number wp of weakly connected components of the digraph Gp is 2. Finally, a conjecture that there are infinitely many primes p such that wp = 2 is presented.

Keywords
connectivity, digraph, prime
Mathematical Subject Classification 2010
Primary: 05C25
Secondary: 05C40, 05C20
Milestones
Received: 2 February 2020
Revised: 10 March 2020
Accepted: 14 March 2020
Published: 30 March 2020

Communicated by Vadim Ponomarenko
Authors
Siji Chen
RDFZ Xishan AP Center
RDFZ Xishan School
Beijing
China
Sheng Chen
Department of Mathematics
Harbin Institute of Technology
Harbin
China