Classifying Heptavalent Symmetric Graphs of Order 40p

A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. Let p be a prime. In this paper, we proved that there is only one connected heptavalent symmetric graphs of order 40p, it is a vertex primitive graph of order 40.3 = 120 admitting S7 as its full automorphism group.