The highest rank of a string C-group representation of the alternating group A_n is known for each n, but no self-dual representations attaining this highest rank are known when n > 12. Motivated by computational results for alternating groups of small degree, we examine a vertex-gluing construction for permutation representation graphs. We establish conditions under which gluing two string C-groups produces another string C-group, and use this construction to obtain infinite families of self-dual representations of alternating groups. In particular, for every n = 4m+3 ≥ 15, we construct ⌊n+9/8⌋ distinct self-dual string C-groups of rank 2m isomorphic to A_n. These representations have rank one below the maximum possible rank of string C-group representations for A_n, and to the authors' knowledge are the highest-rank self-dual representations currently known for alternating groups.
Rank-metric codes, defined as sets of matrices over a finite field with the rank distance, have gained significant attention due to their applications in network coding and connections to diverse mathematical areas. Initially studied by Delsarte in 1978 and later rediscovered by Gabidulin, these codes have become a central topic in coding theory. This paper surveys the development and mathematical foundations, in particular, regarding bounds and constructions of rank-metric codes, emphasizing their extension beyond finite fields to more general settings. We examine Singleton-like bounds on code parameters, demonstrating their sharpness in finite field cases and contrasting this with contexts where the bounds are not tight. Furthermore, we discuss constructions of Maximum Rank Distance (MRD) codes over fields with cyclic Galois extensions and the relationship between linear rank-metric codes with systems and evasive subspaces. The paper also reviews results for algebraically closed fields and real numbers, previously appearing in the context of topology and measure theory. We conclude by proposing future research directions, including conjectures on MRD code existence and the exploration of rank-metric codes over various field extensions.
We classify all regular polyhedra according to their type i.e., the collection of numbers of common neighbours that any pair of distinct vertices may have (polyhedra are planar, 3-connected graphs). As an application, we recover the classification of planar Deza graphs. Next, we focus on the class of quartic polyhedral Deza graphs, and completely characterise it in terms of medial graphs of certain specific cubic polyhedra. Furthermore, within the aforementioned class of quartic polyhedral Deza graphs, we study the extremal graphs with respect to the ratio of number of triangular faces to the total. In the maximal extreme, these notably coincide with the class of line graphs of cubic polyhedra of girth 5. We also fully characterise the quartic polyhedra of type {0,1,2,3}, and in particular we prove that none of them are medial graphs. On one hand our findings fit within the novel research area of common neighbours in graphs. On the other hand, our findings imply general properties of regular planar graphs and regular polyhedra.
Recall that the set of Fubini rankings on n competitors consists of the n-tuples that encode the possible rankings of n competitors in a competition allowing ties. Moreover, recall that a run (weak run) in a tuple is a subsequence of consecutive ascents (weak ascents). If the leading terms of the set of maximally long runs (weak runs) of a tuple are in increasing (weakly increasing) order, then the tuple is said to be flattened (weakly flattened). We define the set of strictly flattened Fubini rankings, which is the subset of Fubini rankings with runs of strict ascents whose leading term are strictly increasing. Analogously, we define the set of weakly flattened Fubini rankings, which is the subset of Fubini rankings with runs of weak ascents whose leading terms are in weakly increasing order. Our main results give formulas for the enumeration of strictly flattened Fubini rankings and weakly flattened Fubini rankings. We also provide some conjectures for further study.
A Catalan word w is said to be flattened if the subsequence of w obtained by taking the first letter of each weakly increasing run is nondecreasing. Let ℱ_n denote the set of flattened Catalan words of length n, which has cardinality 3^n-1+1/2 for all n ≥ 1. In this paper, we consider the distribution of several consecutive patterns on ℱ_n. Indeed, we find explicit formulas for the generating functions of the joint distribution on ℱ_n of several trios of patterns, along with an auxiliary parameter. As special cases of these formulas, we obtain the generating function for the distribution of all consecutive patterns of length two or three. The following equivalences with regard to being identically distributed on ℱ_n arise when comparing the various generating functions and may be explained bijectively: 112≈122 and 211≈221≈231. In addition, explicit expressions are found for the total number of occurrences on ℱ_n of each pattern of length two or three as well as for the number of avoiders of each pattern. These results can be obtained as special cases of our more general formulas for the generating functions, but may be explained combinatorially as well, the arguments of which are featured herein.
We determine the paint cost spectrum for perfect k-ary trees. A coloring of the vertices of a graph G with d colors is said to be d-distinguishing if only the trivial automorphism preserves the color classes. The smallest such d is the distinguishing number of G and is denoted (G). The paint cost of d-distinguishing G, denoted ρ^d(G), is the minimum size of the complement of a color class over all d-distinguishing colorings. A subset S of the vertices of G is said to be a fixing set for G if the only automorphsim that fixes the vertices in S pointwise is the trivial automorphism. The cardinality of a smallest fixing set is denoted (G). In this paper, we explore the breaking of symmetry in perfect k-ary trees by investigating what we define as the paint cost spectrum of a graph G: ((G); ρ^(G)(G), ρ^(G)+1(G), …, ρ^(G)+1(G)) and the paint cost ratio of G, which is defined to be the fraction of paint costs in the paint cost spectrum equal to (G). We determine both the paint cost spectrum and the paint cost ratio completely for perfect k-ary trees. We also prove a lemma that is of interest in its own right: given an n-tuple, n ≥ 2 of distinct elements of an ordered abelian group and 1 ≤ k ≤ n! -1, there exists a k × n row permuted matrix with distinct column sums.
A nut graph is a non-trivial simple graph such that its adjacency matrix has a one-dimensional null space spanned by a full vector. It was recently shown by the authors that there exists a $d$-regular circulant nut graph of order $n$ if and only if $4 \mid d, \, 2 \mid n, \, d>0$, together with $n \ge d + 4$ if $d \equiv_8 4$ and $n \ge d + 6$ if $8 \mid d$, as well as $(n, d) \neq (16, 8)$ [arXiv:2212.03026, 2022]. In this paper, we demonstrate the existence of a $d$-regular Cayley nut graph of order $n$ for each $4 \mid d, \, d>0$ and $2 \mid n, \, n \ge d + 4$, thereby resolving the existence problem for Cayley nut graphs and vertex-transitive nut graphs whose degree is divisible by four.
We investigate Tukey morphisms between binary relations, establishing several fundamental lemmas. We then specialize to finite binary relations, using computational methods to classify all binary relations with at most 6 points in the domain and codomain up to bimorphism. Finally we give a construction of finite binary relations with arbitrary dominating number and dual dominating number.
