Classifying Connected Pentavalent Symmetric Graphs of Order 40pq

A graph is said to be symmetric if the action of its automorphism group on the set of arcs is transitive (arc-transitive). Building upon prior work regarding the classification of symmetric graphs of valency five, this article investigates connected symmetric graphs of order 40pq, where p and q are distinct primes. We establish that the full automorphism group of any such graph is isomorphic to Z5×PSL(2,479), PSL(2,479):D10, or Z5×(PSL(2,479):Z2). Moreover, the vertex stabilizers are shown to be isomorphic to A5 in the first instance and to S5 in the latter two cases.