2-Geodesic-transitive graphs of order twice a prime power

Abstract: In this paper, we study 2-geodesic-transitive graphs of order twice an odd prime power. Classifications of corresponding basic graphs and such graphs with almost simple automorphism groups are given, and a reduction theorem for general case is obtained. Certain new 2-geodesic-transitive graphs are found, and a Magma code regarding 2-geodesic-transitive graphs is provided.

• The paper provides a complete classification of connected basic 2-geodesic-transitive graphs of order 2p^n, delineating cases by automorphism group properties.
• It employs advanced group-theoretic techniques, including O’Nan-Scott theory and computational verification via Magma, to rigorously exclude extraneous cases.
• The study introduces new infinite and sporadic graph families while offering reduction theorems for non-basic cases, setting directions for future research.

Classification of 2-Geodesic-Transitive Graphs of Order Twice a Prime Power

Introduction and Context

The study addresses the structural and classification properties of finite, connected, simple, undirected graphs that are 2-geodesic-transitive and have order $2p^n$ with $p$ an odd prime. A graph is 2-geodesic-transitive if its automorphism group acts transitively on arcs and on vertex triples $(\alpha, \beta, \gamma)$—called 2-geodesics—where $\alpha$ and $\gamma$ are non-adjacent but both adjacent to $\beta$. The 2-geodesic-transitivity property generalizes 2-arc-transitivity, which is central to algebraic graph theory and permutation group analysis.

The paper provides a comprehensive classification of such graphs, generalizes structural reduction theorems for cases with imprimitive or non-quasiprimitive automorphism groups, and investigates the case where the automorphism group is almost simple. The work also identifies new families of 2-geodesic-transitive graphs and provides computational tools for their analysis.

Main Results: Structure and Classification

Classification of Basic Cases

The principal result is a full classification of connected basic (in the sense of permutation group theory) 2-geodesic-transitive graphs of order $2p^n$. The classification is split into two main regimes depending on whether the automorphism group $G$ is quasiprimitive or biquasiprimitive:

(A) The 2-arc-transitive case:

If the graph is also 2-arc-transitive (which can be viewed as a subclass of 2-geodesic-transitive graphs), the possibilities are:

• The Petersen graph ($\mathcal{O}_2$) with automorphism group $S_5$.
• The Hoffman-Singleton graph with automorphism group $p$0.
• Incidence and non-incidence graphs of specific Hadamard designs or projective geometries.
• Complete bipartite graphs $p$1 and related constructions.
• Certain biprimitive normal bi-Cayley graphs over elementary abelian $p$2-groups.

(B) The 2-geodesic-transitive but not 2-arc-transitive case:

Here, the only possibilities for basic graphs are canonical examples built from permutation groups of rank 3, including:

• The Johnson graph $p$3 (order 10, automorphism group $p$4).
• The complement of the Hoffman-Singleton graph and related graphs.
• The orbital graphs of $p$5 on 162 vertices and their complements.
• The graph $p$6, i.e., a complete graph with a perfect matching removed.
• Cartesian products and normal quotients arising from almost simple and product-type automorphism groups.

A complete list of graphs when the automorphism group is almost simple (i.e., between its socle $p$7 and $p$8) is also provided, with precise connection to rank 3 permutation groups and their orbital graphs.

Reduction for Non-Basic Cases

For the non-basic case (where $p$9 is neither quasiprimitive nor biquasiprimitive), the authors provide a reduction theorem: Any such 2-geodesic-transitive graph is either an explicit multipartite complete graph or a normal cover of one of the aforementioned basic cases. This enables a recursive classification of all such graphs via their quotient structure, aligning with classical methods for arc-transitive and s-arc-transitive graphs.

Strong Claims and Computational Aspects

The paper makes strong claims of exhaustiveness in the basic and almost simple cases: there are no other possible basic 2-geodesic-transitive graphs of order $(\alpha, \beta, \gamma)$0 up to isomorphism aside from those in the explicit list. The proof relies on an array of group-theoretic tools (including Guralnick's classification of simple groups with prime-power index subgroups), permutation group O'Nan-Scott theory, and computational calculations validated via Magma code provided in the appendix.

The paper also provides contradictory evidence regarding potentially hypothetical graphs arising from other nonabelian simple group actions (e.g., certain rows in the group-theoretic classification table are ruled out via order or rank arguments).

Implications and Theoretical Significance

Structural Group-Theoretic Insights

The results situate 2-geodesic-transitive graphs within the broader framework of permutation group actions, particularly highlighting the role of almost simple and product action groups. The connection to rank 3 actions reveals a rich interaction between geometric properties of graphs (such as geodesic transitivity) and the combinatorial possibilities of high-rank and multiply transitive group actions.

Computational and Constructive Advances

By providing explicit Magma code, the paper not only enables confirmation of the theoretical classification but also facilitates the discovery of sporadic examples. Such computational tools are essential in verifying structural claims in large or complex cases where manual analysis is infeasible.

New Families and Open Questions

Several new infinite and sporadic families of 2-geodesic-transitive graphs are documented, and the interplay of known and unknown cases is synthesized through the proposal of open problems (e.g., Problem A: determining all 2-geodesic-transitive, but not 2-arc-transitive, graphs with full automorphism group of rank 3 almost simple type). These suggest a fruitful direction for further exploration, particular in the interface with rank 3 permutation group theory and design theory.

Future Directions

The results encourage examination of:

• The existence and properties of as-yet-unclassified rank 3 almost simple groups giving rise to novel 2-geodesic-transitive graphs.
• The combinatorial and spectral invariants of the new graph families discovered.
• Extensions to cases where the order is not twice a prime power or where the girth or other local constraints are modified.
• The adaptation of the proposed classification and reduction techniques to higher-geodesic (e.g., 3-geodesic-transitive) or s-arc-transitive contexts.

Conclusion

This paper conclusively classifies all basic 2-geodesic-transitive graphs of order $(\alpha, \beta, \gamma)$1 for odd primes $(\alpha, \beta, \gamma)$2, categorizes those with (almost) simple automorphism groups, and provides reduction theorems for general graphs in this class. The findings integrate permutation group theory, incidence geometry, and computational group theory, offering a detailed structural picture and new tools for the ongoing analysis of highly symmetric graphs. The implications touch both algebraic graph theory and the theory of permutation groups, guiding the search for new strongly symmetric combinatorial objects and deepening the interplay between group actions and graph geometries.

Reference: "2-Geodesic-transitive graphs of order twice a prime power" (2604.03944)