Diameter 2 Cayley graphs of dihedral groups

We consider the degree-diameter problem for Cayley graphs of dihedral groups. We find upper and lower bounds on the maximum number of vertices of such a graph with diameter 2 and degree d. We completely determine the asymptotic behaviour of this class of graphs by showing that both limits are asymptotically d2/2

Diameter, Girth And Other Properties Of Highly Symmetric Graphs

We consider a number of problems in graph theory, with the unifying theme being the properties of graphs which have a high degree of symmetry. In the degree-diameter problem, we consider the question of finding asymptotically large graphs of given degree and diameter. We improve a number of the current best published results in the case of Cayley graphs of cyclic, dihedral and general groups. In the degree-diameter problem for mixed graphs, we give a new corrected formula for the Moore bound and show non-existence of mixed Cayley graphs of diameter 2 attaining the Moore bound for a range of open cases. In the degree-girth problem, we investigate the graphs of Lazebnik, Ustimenko and Woldar which are the best asymptotic family identified to date. We give new information on the automorphism groups of these graphs, and show that they are more highly symmetrical than has been known to date. We study a related problem in group theory concerning product-free sets in groups, and in particular those groups whose maximal product-free subsets are complete. We take a large step towards a classification of such groups, and find an application to the degree-diameter problem which allows us to improve an asymptotic bound for diameter 2 Cayley graphs of elementary abelian groups. Finally, we study the problem of graphs embedded on surfaces where the induced map is regular and has an automorphism group in a particular family. We give a complete enumeration of all such maps and study their properties

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

Large Networks of Diameter Two Based on Cayley Graphs

In this contribution we present a construction of large networks of diameter two and of order $\frac{1}{2}d^2$ for every degree $d\geq 8$, based on Cayley graphs with surprisingly simple underlying groups. For several small degrees we construct Cayley graphs of diameter two and of order greater than $\frac23$ of Moore bound and we show that Cayley graphs of degrees $d\in\{16,17,18,23,24,31,\dots,35\}$ constructed in this paper are the largest currently known vertex-transitive graphs of diameter two.Comment: 9 pages, Published in Cybernetics and Mathematics Applications in Intelligent System