Related papers: Enumeration of dicirculant digraphs

For a prime $p$ we define the Paley graph to be the graph with the set of vertices $\mathbb{Z}/p\mathbb{Z}$, and with edges connecting vertices whose sum is a quadratic residue. Paley graphs are notoriously difficult to study, particularly…

Using the Polya Enumeration Theorem, we count with particular attention to C^3/Gamma up to C^6/Gamma, abelian orbifolds in various dimensions which are invariant under cycles of the permutation group S_D. This produces a collection of…

We offer a new structural basis for the theory of 3-connected graphs, providing a unique decomposition of every such graph into parts that are either quasi 4-connected, wheels, or thickened $K_{3,m}$'s. Our construction is explicit,…

A finite group $R$ is a DCI-group if, whenever $S$ and $T$ are subsets of $R$ with the Cayley graphs ${\rm Cay}(R,S)$ and ${\rm Cay}(R,T)$ isomorphic, there exists an automorphism $\varphi$ of $R$ with $S^\varphi=T$. Elementary abelian…

We prove that, for any prime $p$, there are precisely $2p^4-p^3-p^2-3p-1$ medial quasigroups of order $p^2$, up to isomorphism.

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field \(K=\mathbb{Q}(\sqrt{d})\), p-ring spaces \(V_p(c)\) modulo c are introduced by defining a morphism \(\psi:\,f\mapsto V_p(f)\) from the divisor…

DP-coloring (also known as correspondence coloring) of a simple graph is a generalization of list coloring. It is known that planar graphs without 4-cycles adjacent to triangles are 4-choosable, and planar graphs without 4-cycles are…

A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of order $s$ has size at least $t$. In 2024, Zhan conjectured that every $2$-connected $[p + 2, p]$-graph of order at least $2p + 3$ and with minimum degree at least $p$…

Abelian Cayley digraphs can be constructed by using a generalization to $Z^n$ of the concept of congruence in $Z$. Here we use this approach to present a family of such digraphs, which, for every fixed value of the degree, have…

A scramble on a connected multigraph is a collection of connected subgraphs that generalizes the notion of a bramble. The maximum order of a scramble, called the scramble number of a graph, was recently developed as a tool for lower…

Let $G$ be a finite group and $\text{cd}(G)$ denote the character degree set for $G$. The prime graph $\Delta(G)$ is a simple graph whose vertex set consists of prime divisors of elements in $\text{cd}(G)$, denoted $\rho(G)$. Two primes…

The dichromatic number of a digraph is the minimum integer $k$ such that it admits a $k$-dicolouring, i.e. a partition of its vertices into $k$ acyclic subdigraphs. We say that a digraph $D$ is a super-orientation of an undirected graph $G$…

This study is the $10^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. In this part, we obtain families of Type-2 isomorphic circulant graphs $C_{np^3}(R)$ w.r.t. $m$ = $p$, and…

A sequence $\sigma$ of $p$ non-negative integers is unigraphic if it is the degree sequence of exactly one graph, up to isomorphism. A polyhedral graph is a $3$-connected, planar graph. We investigate which sequences are unigraphic with…

For two integers $n\geq 3$ and $2\leq p\leq n$, we denote $D(n,p)$ the digraph obtained from a directed $n$-cycle by changing the orientations of $p-1$ consecutive arcs. In this paper, we show that a family of $k$-regular $(k\geq 3)$…

We consider the derangement graph in which the vertices are permutations of $\{ 1,\ldots, n\}$. Two vertices are joined by an edge if the corresponding permutations differ in every position. The derangement graph is known to be Hamiltonian…

The WL-rank of a digraph $\Gamma$ is defined to be the rank of the coherent configuration of $\Gamma$. We construct a new infinite family of strictly Deza Cayley graphs for which the WL-rank is equal to the number of vertices. The graphs…

A Cayley digraph $\Gamma$ over a finite group $G$ is said to be CI if for every Cayley digraph $\Gamma^\prime$ over $G$ isomorphic to $\Gamma$, there is an isomorphism from $\Gamma$ to $\Gamma^\prime$ which is at the same time an…

This is my dissertation about digraphs ordered by pp-constructability. We study in particular smooth digraphs, i.e., digraphs without sources or sinks, tournaments and semicomplete digraphs, orientations of paths and cycles, digraphs with…

Let $\Gamma=$Cay$(G,T)$ be a Cayley digraph over a finite Abelian group $G$ with respect the generating set $T\not\ni0$. $\Gamma$ has order ord$(\Gamma)=|G|=n$ and degree deg$(\Gamma)=|T|=d$. Let $k(\Gamma)$ be the diameter of $\Gamma$ and…