Symmetric graphs of order four times a prime power and valency seven

Abstract

In this paper, we study symmetric graphs of valency seven and order [Formula: see text] with [Formula: see text] a prime and [Formula: see text] an arbitrary positive integer. It is proved that no such graph exists for any prime [Formula: see text], thus reducing the study to the case [Formula: see text]. The result is then used to determine all such graphs of order [Formula: see text], and such graphs of order [Formula: see text] with ‘[Formula: see text]’ can be inductively investigated similarly.