Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

The girth of cubic graphs

We start with an account of the known bounds for n(3,g), the number of vertices in the smallest trivalent graph of girth g,for g 12, including the construction of the smallest known trivalent graph of girth 9. This particular graph has 58 vertices - the 32 known trivalent graphs with 60 vertices are also catalogued and in some cases constructed. We prove the existence of vertex transitive trivalent graphs of arbitrarily high girth using Cayley graphs. The same result is proved for symmetric (that is vertex transitive and edge transitive) graphs, and a family of 2-arctransitive graphs for which the girth is unbounded is exhibited. The excess of trivalent graphs of girth g is shown to be unbounded as a function of g.A lower bound for the number of vertices in the smallest trivalent Cayley graph of girth g is then found for all g = 9, and in each case it is shown that this bound is attained. We also establish an upper bound for the girth of Cayley graphs of subgroups of Aff (p) thegroup of linear transformations of the form x -> ax + b where a,b are members of the field with p elements and a is non-zero. This family contains thesmallest known trivalent graphs of girth 13 and 14, which are exhibited. Lastly a family of 4-arctransitive graphs for which the girth may be unbounded is constructed using "sextets". There is a graph in this family corresponding to each odd prime, and the family splits into several subfamilies depending on the congruency class of this prime modulo 16. The graphs corresponding to the primes congruent to 3,5,11,13modulo 16 are actually 5-arctransitive. The girth of many of these graphs has been computed and graphs with girths up to and including 32 have been found.<p

Applications of the amalgam method to the study of locally projective graphs

Since its birth in 1980 with the seminal paper [Gol80] by Goldschmidt, the amalgam method has proved to be one of the most powerful tools in the modern study of groups, with interesting applications to graphs. Consider a connected graph Γ with a family L of complete subgraphs (called lines) with α ∈ {2,3} vertices each, and possessing a vertex- and edge-transitive group G of automorphisms preserving L. It is assumed that for every vertex x of Γ, there is a bijection between the set of lines containing x and the point-set of a projective GF(2)-space. There is a number of important examples of such locally projective graphs, studied and partly classified by Trofimov, Ivanov and Shpectorov, where both classical and sporadic simple groups appear among the automorphism groups. To a locally projective graph one can associate the corresponding locally projective amalgam A = {G(x),G{l}} comprised of the stabilisers in G of a vertex x and of a line l containing it. The renowned Goldschmidt amalgams turn out to belong to this family (α = 3), as well as their densely embedded Djokovic-Miller subamalgams (α = 2). We first determine all the embeddings of the Djokovic-Miller amalgams in the Goldschmidt amalgams, by designing and implementing an algorithm in GAP and MAGMA. This gives, as a by-product, a list of some finite completions for both the Goldschmidt and the Djokovic-Miller amalgams. Next, we consider two examples of locally projective graphs, special for being devoid of densely embedded subgraphs, and we extend their corresponding locally projective amalgams through the notion of a geometric subgraph. In both cases we find a geometric presentation of the amalgams, which we use to prove the simple connectedness of the corresponding geometry. Finally, we use the Goldschmidt’s lemma to classify, up to isomorphism, certain amalgams related to the Mathieu group M24 and the Held group He, as outlined in [Iva18], and we give an explicit construction of the cocycle whose existence and uniqueness is asserted in [Iva18, Lemma 8.5].Open Acces

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-Biregular Graphs and Orthogonal Polynomials

This thesis is about distance-biregular graphs– when they exist, what algebraic and structural properties they have, and how they arise in extremal problems. We develop a set of necessary conditions for a distance-biregular graph to exist. Using these conditions and a computer, we develop tables of possible parameter sets for distancebiregular graphs. We extend results of Fiol, Garriga, and Yebra characterizing distance-regular graphs to characterizations of distance-biregular graphs, and highlight some new results using these characterizations. We also extend the spectral Moore bounds of Cioaba et al. to semiregular bipartite graphs, and show that distance-biregular graphs arise as extremal examples of graphs meeting the spectral Moore bound