What question did this study set out to answer?

The aim is to classify connected symmetric graphs of order 40pq with specific automorphism groups.

March 5, 2026Open Access

Classifying Connected Pentavalent Symmetric Graphs of Order 40pq

Key Points

The aim is to classify connected symmetric graphs of order 40pq with specific automorphism groups.
Analyzed properties of symmetric graphs
Investigated automorphism groups
Established isomorphism conditions for vertex stabilizers
Identified full automorphism groups as isomorphic to specific group structures
Established vertex stabilizers as isomorphic to A5 or S5 depending on the case
Confirmed that valid graphs maintain transitivity in arc actions

Abstract

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.

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