A pair {p, q} of odd primes is called symmetric if |p ? q| = (p ? 1, q ? 1). The symmetry graph of primes is the graph whose vertices are primes and there is an edge between p and q if and only if {p, q} is a symmetric pair. In this paper we partially answer some questions regarding properties of the symmetry graph of primes stated in a recent article by W. Banks, P. Pollack and C. Pomerance. In particular, we provide an explicit example of a connected component of this graph that has two vertices, and prove that Dickson’s conjecture implies the existence of connected components isomorphic to a given finite connected graph.
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