On an Additive Characterization of a Skew Hadamard (n, n−1/ 2 , n−3 4 )-Difference Set in an Abelian Group

We give a combinatorial proof of an additive characterization of a skew Hadamard (n, n−1 2 , n−3 4 )-difference set in an abelian group G. This research was motivated by the p = 4k + 3 case of Theorem 2.2 of Monico and Elia [4] concerning an additive characterization of quadratic residues in Z p. We then use the known classification of skew (n, n−1 2 , n−3 4 )-difference sets in Z n to give a result for integers n = 4k +3 that strengthens and provides an alternative proof of the p = 4k + 3 case of Theorem 2.2 of [4]

Cyclic Permutations in Doubly-Transitive Groups

Let Ω be a finite set of size n. A cyclic permutation on Ω is a permutation whose cycle decomposition is one cycle of length n. This paper classifies all finite doubly-transitive permutation groups which contain a cyclic permutation. The classification appears in Table 1. We use (G, Ω) for a finite doubly-transitive permutation group G acting on a finite set Ω. For other notation and definitions see the self-contained article Cameron [1]

Single-Change Circular Covering Designs

A single-change circular covering design (scccd) based on the set [v] = {1, . . . ,v} with block size k is an ordered collection of b blocks, B = {B1, . . . ,Bb}, each Bi ⊂ [v], which obey: (1) each block differs from the previous block by a single element, as does the last from the first, and, (2) every pair of [v] is covered by some Bi. The object is to minimize b for a fixed v and k. We present some minimal constructions of scccds for arbitrary v when k = 2 and 3, and for arbitrary k when k+1 ≤ v ≤ 2k. Tight designs are those in which each pair is covered exactly once. Start-Finish arrays are used to construct tight designs when v \u3e 2k; there are 2 non-isomorphic tight designs with (v, k) = (9, 4), and 12 with (v, k) = (10, 4). Some non-existence results for tight designs, and standardized, element-regular, perfect, and column-regular designs are also considered

Constructing and Classifying Neighborhood Anti-Sperner Graphs

For a simple graph G let NG(u) be the (open) neighborhood of vertex u ∈ V (G). Then G is neighborhood anti-Sperner (NAS) if for every u there is a v ∈ V(G)\{u} with NG(u) ⊆NG(v). And a graph H is neighborhood distinct (ND) if every neighborhood is distinct, i.e., if NH(u) ≠ NH(v) when u ≠ v, for all u and v ∈ V(H). In Porter and Yucas [3] a characterization of regular NAS graphs was given: ‘each regular NAS graph can be obtained from a host graph by replacing vertices by null graphs of appropriate sizes, and then joining these null graphs in a prescribed manner’. We extend this characterization to all NAS graphs, and give algorithms to construct all NAS graphs from host ND graphs. Then we find and classify all connected r-regular NAS graphs for r = 0, 1, . . ., 6

Double Arrays, Triple Arrays, and Balanced Grids with \u3cem\u3ev\u3c/em\u3e = \u3cem\u3er\u3c/em\u3e + \u3cem\u3ec\u3c/em\u3e - 1

In Theorem 6.1 of McSorley et al. [3] it was shown that, when v = r+c−1, every triple array TA(v, k, λrr, λcc, k : r × c) is a balanced grid BG(v, k, k : r×c). Here we prove the converse of this Theorem. Our final result is: Let v = r +c−1. Then every triple array is a TA(v, k, c−k, r−k, k : r × c) and every balanced grid is a BG(v, k, k : r × c), and they are equivalent

Rhombic tilings of (n,k)-Ovals, (n,k,λ)-cyclic difference sets, and related topics

Each fixed integer nnhas associated with it ⌊n2⌋ rhombs: ρ1,ρ2,…,ρ⌊n2⌋, where, for each 1≤h≤⌊n2⌋, rhomb ρhρh is a parallelogram with all sides of unit length and with smaller face angle equal toh×πn radians. An Oval is a centro-symmetric convex polygon all of whose sides are of unit length, and each of whose turning angles equalsℓ×πn for some positive integer ℓℓ. A (n,k)(n,k)-Oval is an Oval with 2k2k sides tiled with rhombsρ1,ρ2,…,ρ⌊n2⌋; it is defined by its Turning Angle Index Sequence, a kk-composition of nn. For any fixed pair (n,k)(n,k) we count and generate all (n,k)(n,k)-Ovals up to translations and rotations, and, using multipliers, we count and generate all (n,k)(n,k)-Ovals up to congruency. For odd nn if a (n,k)(n,k)-Oval contains a fixed number λλ of each type of rhombρ1,ρ2,…,ρ⌊n2⌋ then it is called a magic (n,k,λ)(n,k,λ)-Oval. We prove that a magic (n,k,λ)(n,k,λ)-Oval is equivalent to a (n,k,λ)(n,k,λ)-Cyclic Difference Set. For even nn we prove a similar result. Using tables of Cyclic Difference Sets we find all magic (n,k,λ)(n,k,λ)-Ovals up to congruency for n≤40n≤40. Many related topics including lists of (n,k)(n,k)-Ovals, partitions of the regular 2n2n-gon into Ovals, Cyclic Difference Families, partitions of triangle numbers, uu-equivalence of (n,k)(n,k)-Ovals, etc., are also considered

ZEONS, PERMANENTS, THE JOHNSON SCHEME, AND GENERALIZED DERANGEMENTS

Starting with the zero-square “zeon algebra,” the connection with permanents is shown. Permanents of submatrices of a linear combination of the identity matrix and all-ones matrix lead to moment polynomials with respect to the exponential distribution. A permanent trace formula analogous to MacMahon\u27s master theorem is presented and applied. Connections with permutation groups acting on sets and the Johnson association scheme arise. The families of numbers appearing as matrix entries turn out to be related to interesting variations on derangements. These generalized derangements are considered in detail as an illustration of the theory

On \u3cem\u3ek\u3c/em\u3e-minimum and \u3cem\u3em\u3c/em\u3e-minimum Edge-Magic Injections of Graphs

An edge-magic total labelling (EMTL) of a graph G with n vertices and e edges is an injection λ:V(G) ∪ E(G)→[n+e], where, for every edge uv ∈ E(G), we have wtλ(uv)=kλ, the magic sum of λ. An edge-magic injection (EMI) μ of G is an injection μ : V(G) ∪ E(G) → N with magic sum kμ and largest label mμ. For a graph G we define and study the two parameters κ(G): the smallest kμ amongst all EMI’s μ of G, and m(G): the smallest mμ amongst all EMI’s μ of G. We find κ(G) for G ∈ G for many classes of graphs G. We present algorithms which compute the parameters κ(G) and m(G). These algorithms use a G-sequence: a sequence of integers on the vertices of G whose sum on edges is distinct. We find these parameters for all G with up to 7 vertices. We introduce the concept of a double-witness: an EMI μ of G for which both kμ=κ(G) and mμ=m(G) ; and present an algorithm to find all double-witnesses for G. The deficiency of G, def(G), is m(G)−n−e. Two new graphs on 6 vertices with def(G)=1 are presented. A previously studied parameter of G is κEMTL(G), the magic strength of G: the smallest kλ amongst all EMTL’s λ of G. We relate κ(G) to κEMTL(G) for various G, and find a class of graphs B for which κEMTL(G)−κ(G) is a constant multiple of n−4 for G ∈B. We specialise to G=Kn, and find both κ(Kn) and m(Kn) for all n≤11. We relate κ(Kn) and m(Kn) to known functions of n, and give lower bounds for κ(Kn) and m(Kn)