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Let $k,l,m,n$, and $mu$ be positive integers. A $mathbb{Z}_mu$--{it scheme of valency} $(k,l)$ and {it order} $(m,n)$ is a $m times n$ array $(S_{ij})$ of subsets $S_{ij} subseteq mathbb{Z}_mu$ such that for each row and column one has $sum_{j=1}^n |S_{ij}| = k $ and $sum_{i=1}^m |S_{ij}| = l$, respectively. Any such scheme is an algebraic equivalent of a $(k,l)$-semi-regular bipartite voltage graph with $n$ and $m$ vertices in the bipartition sets and voltages coming from the cyclic group $mathbb{Z}_mu$. We are interested in the subclass of $mathbb{Z}_mu$--schemes that are characterized by the property $a - b + c - d; ot equiv ;0$ (mod $mu$) for all $a in S_{ij}$, $b in S_{ih}$, $c in S_{gh}$, and $d in S_{gj}$ where $i,g in {1,...,m}$ and $j,h in {1,...,n}$ need not be distinct. These $mathbb{Z}_mu$--schemes can be used to represent adjacency matrices of regular graphs of girth $ge 5$ and semi-regular bipartite graphs of girth $ge 6$. For suitable $rho, sigma in mathbb{N}$ with $rho k = sigma l$, they also represent incidence matrices for polycyclic $(rho mu_k, sigma mu_l)$ configurations and, in particular, for all known Desarguesian elliptic semiplanes. Partial projective closures yield {it mixed $mathbb{Z}_mu$-schemes}, which allow new constructions for Krv{c}adinacs sporadic configuration of type $(34_6)$ and Balbuenas bipartite $(q-1)$-regular graphs of girth 6 on as few as $2(q^2-q-2)$ vertices, with $q$ ranging over prime powers. Besides some new results, this survey essentially furnishes new proofs in terms of (mixed) $mathbb{Z}_mu$--schemes for ad-hoc constructions used thus far.
We study whether and how can we model a joint distribution $p(x,z)$ using two conditional models $p(x|z)$ and $q(z|x)$ that form a cycle. This is motivated by the observation that deep generative models, in addition to a likelihood model $p(x|z)$, of
Let $triangleleft$ be a relation between graphs. We say a graph $G$ is emph{$triangleleft$-ubiquitous} if whenever $Gamma$ is a graph with $nG triangleleft Gamma$ for all $n in mathbb{N}$, then one also has $aleph_0 G triangleleft Gamma$, where $alph
A graph $G$ is said to be $preceq$-ubiquitous, where $preceq$ is the minor relation between graphs, if whenever $Gamma$ is a graph with $nG preceq Gamma$ for all $n in mathbb{N}$, then one also has $aleph_0 G preceq Gamma$, where $alpha G$ is the dis
A graph $G$ is said to be ubiquitous, if every graph $Gamma$ that contains arbitrarily many disjoint $G$-minors automatically contains infinitely many disjoint $G$-minors. The well-known Ubiquity conjecture of Andreae says that every locally finite g
In a 1983 paper, G. Ramharter asks what are the extremal arrangements for the cyclic analogues of the regular and semi-regular continuants first introduced by T.S. Motzkin and E.G. Straus in 1956. In this paper we answer this question by showing that