Geodesic-transitive graphs with large diameter

Abstract

We review the nearly complete classification project for finite distance-transitive graphs and compile a list of all known graphs. Interestingly, we find that those graphs with diameter larger than 4, apart from a small finite number of exceptions, are geodesic-transitive. Their geodesics exhibit a clear (often geometric) structure. On the other hand, we provide examples of graphs that are distance-transitive but not geodesic-transitive, including two infinite families with diameter 3 and a few sporadic ones with diameter 3, 4 or 7. In the last section, we extend our investigation to polar Grassmann graphs and provide an explicit description of their geodesics.

Figures (2)

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Theorems & Definitions (34)

• Theorem 1.2
• Proposition 1.3
• Theorem 1.4
• Lemma 2.1
• Proposition 2.2
• Lemma 2.3
• Example 2.4
• Example 2.5
• Lemma 3.1
• proof