[Paper Review] On distance, geodesic and arc transitivity of graphs

Alice Devillers, Wei Jin+2 more|arXiv (Cornell University)|Oct 11, 2011

Finite Group Theory Research18 references3 citations

TL;DR

This paper investigates the hierarchy of symmetry properties in finite graphs—specifically s-distance transitivity, s-geodesic transitivity, and s-arc transitivity—proving that for s ∈ {1,2,3}, there are infinitely many geodesic transitive graphs that are s-arc transitive but not (s+1)-arc transitive, with arbitrarily large diameter. It further classifies all 2-geodesic transitive graphs of prime valency that are not 2-arc transitive, showing they exist if and only if p ≡ 1 (mod 4), and are uniquely realizable as antipodal double covers of K_{p+1} with automorphism group PSL(2,p) × Z₂.

ABSTRACT

We compare three transitivity properties of finite graphs, namely, for a positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and $s$-arc transitivity. It is known that if a finite graph is $s$-arc transitive but not $(s+1)$-arc transitive then $s\leq 7$ and $s eq 6$. We show that there are infinitely many geodesic transitive graphs with this property for each of these values of $s$, and that these graphs can have arbitrarily large diameter if and only if $1\leq s\leq 3$. Moreover, for a prime $p$ we prove that there exists a graph of valency $p$ that is 2-geodesic transitive but not 2-arc transitive if and only if $p\equiv 1\pmod 4$, and for each such prime there is a unique graph with this property: it is an antipodal double cover of the complete graph $K_{p+1}$ and is geodesic transitive with automorphism group $PSL(2,p) imes Z_2$.

Motivation & Objective

• To clarify the relationships between s-distance transitivity, s-geodesic transitivity, and s-arc transitivity in finite graphs.
• To determine for which s there exist geodesic transitive graphs that are s-arc transitive but not (s+1)-arc transitive, with arbitrarily large diameter.
• To classify all 2-geodesic transitive graphs of prime valency that are not 2-arc transitive.
• To identify conditions under which diameter 2 distance transitive graphs fail to be geodesic transitive.
• To explore whether similar classification results can be extended to non-prime valencies.

Proposed method

• The authors analyze well-known families of distance transitive graphs—Johnson graphs, Hamming graphs, Odd graphs, and classical generalized polygons—to establish existence of infinite families of geodesic transitive graphs with desired arc transitivity properties.
• They use group-theoretic constructions, particularly coset graphs of the form Cos(G, H, HgH), to build and analyze graphs with specific symmetry and valency properties.
• For prime valency p ≡ 1 (mod 4), they construct a unique 2-geodesic transitive graph as a nonbipartite antipodal double cover of K_{p+1} using PSL(2,p) and a central involution.
• They prove isomorphism invariance of the construction by showing that choices of subgroups and elements yield isomorphic graphs via conjugation in PGL(2,p).
• They verify 2-geodesic transitivity by analyzing orbit structures of vertex stabilizers and their actions on neighborhoods and second neighborhoods.
• They use known results on automorphism groups and girth to show that the constructed graphs are not 2-arc transitive, relying on the girth being 3 and disjoint neighborhoods.

Experimental results

Research questions

• RQ1 For which values of s do there exist infinitely many geodesic transitive graphs that are s-arc transitive but not (s+1)-arc transitive, and when can such graphs have arbitrarily large diameter?
• RQ2 What are the necessary and sufficient conditions for a 2-geodesic transitive graph of prime valency to fail to be 2-arc transitive?
• RQ3 Is there a unique 2-geodesic transitive graph of prime valency p that is not 2-arc transitive, and what is its automorphism group?
• RQ4 Which diameter 2 distance transitive graphs are not geodesic transitive, and what is the smallest such example?
• RQ5 Can the classification of 2-geodesic transitive graphs of prime valency be extended to non-prime valencies?

Key findings

• For each s ∈ {1,2,3,4,5,7}, there exist infinitely many geodesic transitive graphs that are s-arc transitive but not (s+1)-arc transitive.
• Geodesic transitive graphs that are s-arc transitive but not (s+1)-arc transitive can have arbitrarily large diameter if and only if s ≤ 3.
• For prime valency p, a 2-geodesic transitive graph that is not 2-arc transitive exists if and only if p ≡ 1 (mod 4).
• When such a graph exists, it is uniquely realizable as a nonbipartite antipodal double cover of K_{p+1} with automorphism group PSL(2,p) × Z₂.
• The Paley graph P(q) is distance transitive for all q ≡ 1 (mod 4), but is geodesic transitive only when q = 5 or 9, making it the first known family of diameter 2 graphs that are distance transitive but not geodesic transitive for q > 9.
• The constructed graphs in the classification are 2-geodesic transitive but not 2-arc transitive, with girth 3 and diameter 3, and they admit a unique system of imprimitivity of blocks of size 2.