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

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)

Multivariate Matching Polynomials of Cyclically Labelled Graphs

We consider the matching polynomials of graphs whose edges have been cyclically labelled with the ordered set of t labels {x1, . . ., xt}. We first work with the cyclically labelled path, with first edge label xi, followed by N full cycles of labels {x1, . . ., xt}, and last edge label xj . Let Φi,Nt+j denote the matching polynomial of this path. It satisfies the (τ, Δ)-recurrence: Φi,Nt+j = τΦi,(N−1)t+j−ΔΦi,(N−2)t+j, where τ is the sum of all non-consecutive cyclic monomials in the variables {x1, . . ., xt} and Δ = (−1)t x1 · · ·xt. A combinatorial/algebraic proof and a matrix proof of this fact are given. Let GN denote the first fundamental solution to the (τ, Δ)-recurrence. We express GN (i) as a cyclic binomial using the Symmetric Representation of a matrix, (ii) in terms of Chebyshev polynomials of the second kind in the variables τ and Δ, and (iii) as a quotient of two matching polynomials. We extend our results from paths to cycles and rooted trees

Complete Enumeration and Properties of Binary Pseudo-Youden Designs PYD(9, 6, 6)

A binary pseudo -Youden design PYD(9, 6, 6) is a 6 × 6 array in which each cell contains one element from the set V = {1, 2, . . ., 9}, and each element from V occurs 4 times. Every row of the array contains distinct elements and every column contains distinct elements. The rows and columns, when taken together, are pairwise balanced and form a (9, 12, 8, 6, 5)-BIBD. In Preece (1968) and (1976) a total of 345 species of binary PYD(9, 6, 6) were found. Here we complete this enumeration and find 348 species of binary PYD(9, 6, 6). We give a complete set of invariants for these species based upon the numbers of intercalates and anti-intercalates that they contain; and discuss some of their properties. We also show that there are 696 non-isomorphic binary PYD(9, 6, 6), and give a complete set of invariants for these arrays

Generating Sequences of Clique-Symmetric Graphs via Eulerian Digraphs

Let {Gp1,Gp2, . . .} be an infinite sequence of graphs with Gpn having pn vertices. This sequence is called Kp-removable if Gp1 ≅ Kp, and Gpn − S ≅ Gp(n−1) for every n ≥ 2 and every vertex subset S of Gpn that induces a Kp. Each graph in such a sequence has a high degree of symmetry: every way of removing the vertices of any fixed number of disjoint Kp’s yields the same subgraph. Here we construct such sequences using componentwise Eulerian digraphs as generators. The case in which each Gpn is regular is also studied, where Cayley digraphs based on a finite group are used

A Combinatorial Interpretation of Lommel Polynomials and Their Derivatives

In this paper we present interpretations of Lommel polynomials and their derivatives. A combinatorial interpretation uses matchings in graphs. This gives an interpretation for the derivatives as well. Then Lommel polynomials are considered from the point of view of operator calculus. A step-3 nilpotent Lie algebra and finite-difference operators arise in the analysis