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Metacirculants are a rich resource of many families of interesting graphs, and weak metacirculants are generalizations of them. A graph is called a {em split weak metacirculant} if it has a vertex-transitive split metacyclic automorphism group. In two recent papers, it is shown that a graph of prime power order is a metacirculant if and only if it is a split weak metacirculant. Let $m$ is a positive integer. In this paper, we first give a sufficient condition for the existence of split weak metacirculants of order $m$ which are not metacirculants. This is then used to give a sufficient and necessary condition for the existence of split weak metacirculants of order $n$ which are not metacirculants, where $n$ is a product of two prime-powers. As byproducts, we construct infinitely many split weak metacirculant graphs which are not metacirculant graphs, and answer an open question reported in the literature.
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Let $m,n,r$ be positive integers, and let $G=langle arangle: langle brangle cong mathbb{Z}_n: mathbb{Z}_m$ be a split metacyclic group such that $b^{-1}ab=a^r$. We say that $G$ is {em absolutely split with respect to $langle arangle$} provided that for any $xin G$, if $langle xranglecaplangle arangle=1$, then there exists $yin G$ such that $xinlangle yrangle$ and $G=langle arangle: langle yrangle$. In this paper, we give a sufficient and necessary condition for the group $G$ being absolutely split. This generalizes a result of Sanming Zhou and the second author in [arXiv: 1611.06264v1]. We also use this result to investigate the relationship between metacirculants and weak metacirculants. Metacirculants were introduced by Alspach and Parsons in $1982$ and have been a rich source of various topics since then. As a generalization of this classes of graphs, Maruv siv c and v Sparl in 2008 posed the so called weak metacirculants. A graph is called a {em weak metacirculant} if it has a vertex-transitive metacyclic automorphism group. In this paper, it is proved that a weak metacirculant of $2$-power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. This provides a partial answer to an open question in the literature.
Let $G_{m,n,k} = mathbb{Z}_m ltimes_k mathbb{Z}_n$ be the split metacyclic group, where $k$ is a unit modulo $n$. We derive an upper bound for the diameter of $G_{m,n,k}$ using an arithmetic parameter called the textit{weight}, which depends on $n$, $k$, and the order of $k$. As an application, we show how this would determine a bound on the diameter of an arbitrary metacyclic group.
Motivated by Yabes classification of symmetric $2$-generated axial algebras of Monster type, we introduce a large class of algebras of Monster type $(alpha, frac{1}{2})$, generalising Yabes $mathrm{III}(alpha,frac{1}{2}, delta)$ family. Our algebras bear a striking similarity with Jordan spin factor algebras with the difference being that we asymmetrically split the identity as a sum of two idempotents. We investigate the properties of this algebra, including the existence of a Frobenius form and ideals. In the $2$-generated case, where our algebra is isomorphic to one of Yabes examples, we use our new viewpoint to identify the axet, that is, the closure of the two generating axes.
A split graph is a graph whose vertices can be partitioned into a clique and a stable set. We investigate the combinatorial species of split graphs, providing species-theoretic generalizations of enumerative results due to Bina and Pv{r}ibil (2015), Cheng, Collins, and Trenk (2016), and Collins and Trenk (2018). In both the labeled and unlabeled cases, we give asymptotic results on the number of split graphs, of unbalanced split graphs, and of bicolored graphs, including proving the conjecture of Cheng, Collins, and Trenk (2016) that almost all split graphs are balanced.
Given a strictly increasing sequence $mathbf{t}$ with entries from $[n]:={1,ldots,n}$, a parking completion is a sequence $mathbf{c}$ with $|mathbf{t}|+|mathbf{c}|=n$ and $|{tin mathbf{t}mid tle i}|+|{cin mathbf{c}mid cle i}|ge i$ for all $i$ in $[n]$. We can think of $mathbf{t}$ as a list of spots already taken in a street with $n$ parking spots and $mathbf{c}$ as a list of parking preferences where the $i$-th car attempts to park in the $c_i$-th spot and if not available then proceeds up the street to find the next available spot, if any. A parking completion corresponds to a set of preferences $mathbf{c}$ where all cars park. We relate parking completions to enumerating restricted lattice paths and give formulas for both the ordered and unordered variations of the problem by use of a pair of operations termed textbf{Join} and textbf{Split}. Our results give a new volume formula for most Pitman-Stanley polytopes, and enumerate the signature parking functions of Ceballos and Gonzalez DLeon.
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