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

Dva-ločno-tranzitivni dvo-valentni digrafi določenih redov

The topic of this paper is digraphs of in-valence and out-valence 2 that admit a 2-arc-transitive group of automorphisms. We classify such digraphs that satisfy certain additional conditions on their order. In particular, a classification of those with order ▫$kp$▫ or ▫$kp^{2}$▫ where ▫$k leq 14$▫ and ▫$p$▫ is a prime can be deduced from the results of this paper.Tema tega članka so digrafi vhodne in izhodne valence 2, ki dopuščajo 2-ločno-tranzitivno grupo avtomorfizmov. Klasificiramo takšne digrafe, ki zadoščajo določenim dodatnim pogojem glede njihovega reda. Tako je npr. mogoče s pomočjo rezultatov tega članka klasificirati tiste, ki imajo red ▫$kp$▫ ali ▫$kp^{2}$▫, kjer je ▫$k leq 14$▫ in je ▫$p$▫ praštevilo

Cometric Association Schemes

The combinatorial objects known as association schemes arise in group theory, extremal graph theory, coding theory, the design of experiments, and even quantum information theory. One may think of a d-class association scheme as a (d + 1)-dimensional matrix algebra over R closed under entrywise products. In this context, an imprimitive scheme is one which admits a subalgebra of block matrices, also closed under the entrywise product. Such systems of imprimitivity provide us with quotient schemes, smaller association schemes which are often easier to understand, providing useful information about the structure of the larger scheme. One important property of any association scheme is that we may find a basis of d + 1 idempotent matrices for our algebra. A cometric scheme is one whose idempotent basis may be ordered E0, E1, . . . , Ed so that there exists polynomials f0, f1, . . . , fd with fi ◦ (E1) = Ei and deg(fi) = i for each i. Imprimitive cometric schemes relate closely to t-distance sets, sets of unit vectors with only t distinct angles, such as equiangular lines and mutually unbiased bases. Throughout this thesis we are primarily interested in three distinct goals: building new examples of cometric association schemes, drawing connections between cometric association schemes and other objects either combinatorial or geometric, and finding new realizability conditions on feasible parameter sets — using these conditions to rule out open parameter sets when possible. After introducing association schemes with relevant terminology and definitions, this thesis focuses on a few recent results regarding cometric schemes with small d. We begin by examining the matrix algebra of any such scheme, first looking for low rank positive semidefinite matrices with few distinct entries and later establishing new conditions on realizable parameter sets. We then focus on certain imprimitive examples of both 3- and 4-class cometric association schemes, generating new examples of the former while building realizability conditions for both. In each case, we examine the related t-distance sets, giving conditions which work towards equivalence; in the case of 3-class Q-antipodal schemes, an equivalence is established. We conclude by partially extending a result of Brouwer and Koolen concerning the connectivity of graphs arising from metric association schemes

Hypermaps: constructions and operations

It is conjectured that given positive integers l, m, n with l-1 + m-1 + n-1 < 1and an integer g ≥ 0, the triangle group Δ = Δ (l, m, n) = ⟨X,Y,Z|X l = Y m =Z n = X Y Z = 1⟩ contains infinitely many subgroups of finite index and of genusg. This conjecture can be rewritten in another form: given positive integers l,m, n with l¡1 +m¡1 +n¡1 < 1 and an integer g ≥ 0, there are infinitely manynonisomorphic compact orientable hypermaps of type (l, m, n) and genus g.We prove that the conjecture is true, when two of the parameters l, m, n areequal, by showing how to construct those hypermaps, and we extend the resultto nonorientable hypermaps.A classification of all operations of finite order in oriented hypermaps isgiven, and a detailed study of one of these operations (the duality operation)is developed. Adapting the notion of chirality group, the duality group ofH can be defined as the minimal subgroup D(H) ≤¦ M on (H) such thatH = D (H) is a self-dual hypermap. We prove that for any positive integer d,we can find a hypermap of that duality index (the order of D (H) ), even whensome restrictions apply, and also that, for any positive integer k, we can find anon self-dual hypermap such that |Mon (H) | = d = k. We call this k the dualitycoindex of the hypermap. Links between duality index, type and genus of aorientably regular hypermap are explored.Finally, we generalize the duality operation for nonorientable regular hypermaps and we verify if the results about duality index, obtained for orientably regular hypermaps, are still valid