Related papers: A census of small Schurian association schemes

Let $p$ be an odd prime. In this paper we provide a construction which gives four non-Schurian association schemes for every $p\geq 5$ and two for $p=3$. This construction is explained using incidences between points and lines of a biaffine…

The main result here is a characterisation of binary $2$-neighbour-transitive codes with minimum distance at least $5$ via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of…

The minimal degree of a permutation group $G$ is defined as the minimal number of non-fixed points of a non-trivial element of $G$. In this paper we show that if $G$ is a transitive permutation group of degree $n$ having no non-trivial…

Graphs are commonly used to represent and visualize causal relations. For a small number of variables, this approach provides a succinct and clear view of the scenario at hand. As the number of variables under study increases, the graphical…

A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a permutation group on the set $G$ containing the regular subgroup of all right translations. It was proved by R. P\"oschel (1974) that…

Let $d(G)$ be the minimum number of elements required to generated a group $G.$ For a group $G $ of order $p^n$ with derived subgroup of order $ p^k $ and $d(G) = d,$ we knew the order of the Schur multiplier of $G$ is bounded by $…

In this article we give a classification of the sub-groups in PSL(2,Z) and of the conjugacy classes of these sub-groups by the mean of an combinatorial invariant: some trivalent diagrams (dotted or not). We give explicit formulae enabling…

Using the fact that Hopf-Galois structures on separable extensions and skew bracoids are both intrinsically connected to transitive subgroups of the holomorph of a finite group, we present algorithms to classify and enumerate these objects…

Let $G$ be a finite $p$-group of order $p^n$ and $M(G)$ be its Schur multiplier. It is well known result by Green that $|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}$ for some $t(G) \geq 0$. In this article we classify non-abelian $p$-groups $G$ of…

We give an explicit characterization of solvable factors in factorizations of finite classical groups of Lie type. This completes the classification of solvable factors in factorizations of almost simple groups, finishing the program…

In this paper, for any prime $p$, we propose the notion of a $p$-transitive association scheme. This notion aims to generalize the fact that the regular module of a group algebra of a finite group has a unique trivial submodule to the case…

In this work we employ a simple pairing interaction model in order to study and classify an eventual pairing phase transition in finite fermionic systems. We show that systems with as few as 10-16 fermions can exhibit clear features…

Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The…

A vertex triple $(u,v,w)$ of a graph is called a $2$-geodesic if $v$ is adjacent to both $u$ and $w$ and $u$ is not adjacent to $w$. A graph is said to be $2$-geodesic transitive if its automorphism group is transitive on the set of…

Generalised degrees provide a natural bridge between local and global topological properties of networks. We define the generalised degree to be the number of neighbours of a node within one and two steps respectively. Tailored random graph…

The additive closedness in the subset of an additive group is termed as r-value. The nature of closedness in different subsets of fixed size is observed as a spectrum of r-values. We enumerate r-values of subsets in finite fields of…

We consider a code to be a subset of the vertex set of a Hamming graph. The set of $s$-neighbours of a code is the set of vertices, not in the code, at distance $s$ from some codeword, but not distance less than $s$ from any codeword. A…

Real-world networks often exhibit strong transitivity with nontrivial local clustering spectra and degree correlations. Such features are not easily modeled in tractable network models, creating an obstacle to the theoretical understanding…

In this paper, we obtain classification results for higher-dimensional analogues of classical association schemes called association schemes on triples (ASTs). We present an algorithm that enumerates all ASTs on a fixed number of vertices…

We use character theory and character estimates to show that the number of $T_2$-systems of $A_n$ is at least \begin{equation*} \frac{1}{8n\sqrt{3}}\exp\left(\frac{2\pi}{\sqrt{6}}n^{1/2}\right)(1+o(1)). \end{equation*} Applying this result,…