Asymptotically approaching the Moore bound for diameter three by Cayley graphs

The largest order n(d,k) of a graph of maximum degree d and diameter k cannot exceed the Moore bound, of the form M(d,k) = dk - O(dk-1) for d → ∞ and any fixed k. Known results in finite geometries on generalised (k+1)-gons imply, for k=2,3,5, existence of an infinite sequence of values of d such that n(d,k) = dk - o(dk). Thus, for k = 2,3,5 the Moore bound can be asymptotically approached in the sense that lim supd→ ∞ n(d,k)/M(d,k) =1; moreover, no such result is known for any other value of k ≥ 2. The corresponding graphs are, however, far from vertex-transitive, and there appears to be no obvious way to extend them to vertex-transitive graphs giving the same type of asymptotic result. The second and the third author (2012) proved by a direct construction that the Moore bound for diameter k = 2 can, in a similar sense, be asymptotically approached by Cayley graphs. Subsequently, the first and the third author (2015) showed that the same construction can be derived from generalised triangles with polarity. By a detailed analysis of regular orbits of suitable groups of automorphisms of graphs arising from polarity quotients of incidence graphs of generalised quadrangles with polarity, we prove that for an infinite set of values of d there exist Cayley graphs of degree d, diameter 3, and order d3 - O(d2.5). The Moore bound for diameter 3 can thus as well be asymptotically approached by Cayley graphs. We also show that this method does not extend to constructing Cayley graphs of diameter 5 from generalised hexagons with polarity

Coverings of generalized Petersen graphs

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Kolja Knauer[en] In this project we study the coverings of Generalized Petersen Graphs that are Generalized Petersen Graphs themselves. We give a large family of such Generalized Petersen Graphs and review results of Krnc and Pisanski by focusing on Kronecker double covers. Finally, we generalize their results partially to Kronecker triple covers

Intriguing sets of partial quadrangles

The point-line geometry known as a \textit{partial quadrangle} (introduced by Cameron in 1975) has the property that for every point/line non-incident pair $(P,\ell)$, there is at most one line through $P$ concurrent with $\ell$. So in particular, the well-studied objects known as \textit{generalised quadrangles} are each partial quadrangles. An \textit{intriguing set} of a generalised quadrangle is a set of points which induces an equitable partition of size two of the underlying strongly regular graph. We extend the theory of intriguing sets of generalised quadrangles by Bamberg, Law and Penttila to partial quadrangles, which surprisingly gives insight into the structure of hemisystems and other intriguing sets of generalised quadrangles