HS-integral and Eisenstein integral normal mixed Cayley graphs

New results on the degree/diameter problem of mixed Abelian Cayley graphs

Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case in which such graphs are Cayley graphs of Abelian groups. These groups can be constructed by using a generalization to $\mathbb{Z}^n$ of the concept of congruence in $\mathbb{Z}$. Here we use this approach to present some families of mixed graphs, which, for every fixed value of the degree, have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.

Mixed Moore Cayley graphs

The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected edges and directed arcs in the graph. For a diameter 2 graph with maximum undirected degree $r$ and directed out-degree $z$, a straightforward counting argument yields an upper bound $M(z,r,2)=(z+r)^2+z+1$ for the order of the graph. Apart from the case $r=1$, the only three known examples of mixed graphs attaining this bound are Cayley graphs, and there ar…

An improved Moore bound and some new optimal families of mixed Abelian Cayley graphs

We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first show these bounds can be improved if we know more details about the order of some elements of the generating set. Based on these improvements, we present some new families of mixed graphs. For every fixed value of the degree, these families have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are show…

利用混合二面体群构造正规 Cayley 图，以及一个新的非 Cayley 图的二部 2-弧传递图

Using mixed dihedral groups to construct normal Cayley graphs, and a new bipartite $2$-arc-transitive graph which is not a Cayley graph

本文提出混合二面体群上 Cayley 图自同构群的判定条件，并据此构造 K_{16,16} 的阶 2^{53} 的非 Cayley 2-弧传递正规覆盖，解决了 2009 年的相关公开问题。

H-integral and Gaussian integral normal mixed Cayley graphs

If all the eigenvalues of the Hermitian-adjacency matrix of a mixed graph are integers, then the mixed graph is called \emph{H-integral}. If all the eigenvalues of the (0,1)-adjacency matrix of a mixed graph are \emph{Gaussian integers}, then the mixed graph is called \emph{Gaussian integral}. For any finite group $Γ$, we characterize the set $S$ for which the normal mixed Cayley graph $\text{Cay}(Γ, S)$ is H-integral. We further prove that a normal mixed Cayley graph is H-integral if and only if it is Gaussian integral.

Multiplicities of degenerations of matrices and mixed volumes of Cayley polyhedra

Using a certain Pick-type formula for the mixed volume of Cayley polyhedra, we compute the multiplicity of the isolated common zero of the maximal minors for a matrix of generic homogeneous polynomials of given degrees.

Splitting fields of mixed Cayley graphs over abelian groups

The splitting field $\mathbb{SF}(Γ)$ of a mixed graph $Γ$ is the smallest field extension of $\mathbb{Q}$ which contains all eigenvalues of the Hermitian adjacency matrix of $Γ$. The extension degree $[\mathbb{SF}(Γ):\mathbb{Q}]$ is called the algebraic degree of $Γ$. In this paper, we determine the splitting fields and algebraic degrees of mixed Cayley graphs over abelian groups. This generalizes the main results of [K. Mönius, Splitting fields of spectra of circulant graphs, J. Algebra 594(15) (2022) 154--169] and [M. Kadyan, B. Bhattacharjya, Integral mixed Cayley graphs over abelian groups…

Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups

In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to th…

A characterisation of edge-affine $2$-arc-transitive covers of $\K_{2^n,2^n}$

We introduce the notion of an \emph{$n$-dimensional mixed dihedral group}, a general class of groups for which we give a graph theoretic characterisation. In particular, if $H$ is an $n$-dimensional mixed dihedral group then the we construct an edge-transitive Cayley graph $Γ$ of $H$ such that the clique graph $Σ$ of $Γ$ is a $2$-arc-transitive normal cover of $\K_{2^n,2^n}$, with a subgroup of $\Aut(Σ)$ inducing a particular \emph{edge-affine} action on $\K_{2^n,2^n}$. Conversely, we prove that if $Σ$ is a $2$-arc-transitive normal cover of $\K_{2^n,2^n}$, with a subgroup of $\Aut(Σ)$ inducin…

关于大型正则 (1,1,k)-混合图

On large regular (1,1,k)-mixed graphs

本文研究正则 $(1,1,k)$-混合图的最大顶点数问题，结合计算机搜索与字母表图方法，构造了 Cayley 图、提升图及直径约为 $2\log_2 N$ 的无限族。

A family of $2$-groups and an associated family of semisymmetric, locally $2$-arc-transitive graphs

A mixed dihedral group is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper, for each $n\geq 2$, we construct a mixed dihedral $2$-group $H$ of nilpotency class $3$ and order $2^a$ where $a=(n^3+n^2+4n)/2$, and a corresponding graph $Σ$, which is the clique graph of a Cayley graph of $H$. We prove that $Σ$ is semisymmetric, that is, ${\rm Aut}(Σ)$ acts transitively on the edges, but intransitively on the vertices, of $Σ$. These graphs are the first known semisymmetric gr…