A note on pentavalent s-transitive graphs

AbstractA graph, with a group G of its automorphisms, is said to be (G,s)-transitive if G is transitive on s-arcs but not on (s+1)-arcs of the graph. Let X be a connected (G,s)-transitive graph for some s≥1, and let Gv be the stabilizer of a vertex v∈V(X) in G. In this paper, we determine the structure of Gv when X has valency 5 and Gv is non-solvable. Together with the results of Zhou and Feng [J.-X. Zhou, Y.-Q. Feng, On symmetric graphs of valency five, Discrete Math. 310 (2010) 1725–1732], the structure of Gv is completely determined when X has valency 5. For valency 3 or 4, the structure of Gv is known

The heptagon-wheel cocycle in the Kontsevich graph complex

The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on $n$ vertices and $2n-2$ edges, induce -- under the orientation mapping -- infinitesimal symmetries of classical Poisson structures on arbitrary finite-dimensional affine real manifolds. Willwacher has stated the existence of a nontrivial cocycle that contains the $(2\ell+1)$-wheel graph with a nonzero coefficient at every $\ell\in\mathbb{N}$. We present detailed calculations of the differential of graphs; for the tetrahedron and pentagon-wheel cocycles, consisting at $\ell = 1$ and $\ell = 2$ of one and two graphs respectively, the cocycle condition $d(\gamma) = 0$ is verified by hand. For the next, heptagon-wheel cocycle (known to exist at $\ell = 3$), we provide an explicit representative: it consists of 46 graphs on 8 vertices and 14 edges.Comment: Special Issue JNMP 2017 `Local and nonlocal symmetries in Mathematical Physics'; 17 journal-style pages, 54 figures, 3 tables; v2 accepte

Distance-regular Cayley graphs with small valency

We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most $4$, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency $5$, and the Cayley graphs among all distance-regular graphs with girth $3$ and valency $6$ or $7$. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some ``exceptional'' distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs.Comment: 19 pages, 4 table