Degree/diameter problem for mixed graphs

The Degree/diameter problem asks for the largest graphs given diameter and maximum degree. This problem has been extensively studied both for directed and undirected graphs, ando also for special classes of graphs. In this work we present the state of art of the degree/diameter problem for mixed graphs

The diameter of random Cayley digraphs of given degree

We consider random Cayley digraphs of order $n$ with uniformly distributed generating set of size $k$. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as $n\to\infty$ and $k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n \log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the probability converges exponentially fast to 1.Comment: 11 page

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