Finite 2-geodesic transitive graphs of valency \(3p\). (Q2810177)

A \(2\)-geodesic is an ordered triple of vertices \((u,v,w)\) with \(u\) adjacent to \(v\), \(v\) adjacent to \(w\), but \(u\) neither equal nor adjacent to \(w\) (i.e., \(u\) and \(w\) being of distance \(2\)). In other words, a \(2\)-geodesic is a \(2\)-arc whose starting and ending points are non-adjacent. A graph is said to be \(2\)-geodesic transitive if it admits an automorphism group acting transitively on its set of \(2\)-geodesics. A \(2\)-arc transitive graph is necessarily \(2\)-geodesic transitive, but if a graph contains \(2\)-geodesics as well as \(2\)-arcs whose end-points are adjacent (in which case the graph contains triangles and its girth must be equal to \(3\)), it cannot be \(2\)-arc transitive while it still can be \(2\)-geodesic transitive (with the \(4\)-partite \(K_{4[3]}\) being a prime example).NEWLINENEWLINE All \(2\)-geodesic transitive graphs of valency \(4\), prime valency, as well as all \(2\)-geodesic transitive graphs of valency equal to twice a prime have been classified. This paper claims to provide a classification of all \(2\)-geodesic transitive graphs of valency \(3p\), \(p\) an odd prime, but what it really contains is a division of all such graphs into four classes based on the action of the vertex stabilizer and a classification of two of these classes. Namely, the author shows that a \(2\)-geodesic graph of valency \(3p\) with a connected vertex-neighborhood and the vertex stabilizer acting primitively on the vertex-neighborhood must be one from a list of \(10\) graphs, while a \(2\)-geodesic graph of valency \(3p\) with a connected vertex-neighborhood and the vertex stabilizer acting transitively but imprimitively must be the \((m+1)\)-partite \( K_{(m+1)[n]} \) with \((m,n)\) either equal to \((3,p)\) or \((p,3)\). If the vertex-neighborhood is connected but the stabilizer does not act transitively on it, the graph induced by the vertex-neighborhood must be of diameter \(2\), and such graphs exist for all odd primes \(p\) (with Johnson graphs serving as examples). If the vertex-neighborhood is disconnected, it must be isomorphic to \(mK_n\) with \((m,n)\) equal to \((3,p)\) or \((p,3)\). Finally, a \(2\)-geodesic graph of valency \(3p\) may be \(2\)-arc transitive (of girth at least \(4\)). The proofs are not particularly hard and rely on some good structural insights and previously proved results.