Related papers: Circulant Digraphs Integral over Number Fields

From a generalization to $Z^n$ of the concept of congruence we define a family of regular digraphs or graphs called multidimensional circulants, which turn out to be Cayley (di)graphs of Abelian groups. This paper is mainly devoted to show…

If $\Gamma$ is a graph for which every edge is in exactly one clique of order $\omega$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $\Gamma$. We discover many general…

Integral circulant graphs are proposed as models for quantum spin networks that permit a quantum phenomenon called perfect state transfer. Specifically, it is important to know how far information can potentially be transferred between…

Let K be a number field. A finite group G is called K-admissible if there exists a G-crossed product K-division algebra. K-admissibility has a necessary condition called K-preadmissibility that is known to be sufficient in many cases. It is…

We consider the problem of defining polynomials over function fields of positive characteristic. Among other results, we show that the following assertions are true. 1. Let $\G_p$ be an algebraic extension of a field of $p$ elements and…

Let $S\subseteq R$ be a multiplicatively closed subset of a ring $R$. We extend several results on integral domains to their $S$-versions and establish the $S$-version of Krull intersection theorem. We also show that if $R$ is an $S$-field,…

We introduce a new model for investigating spectral properties of quantum graphs, a quantum circulant graph. Circulant graphs are the Cayley graphs of cyclic groups. Quantum circulant graphs with standard vertex conditions maintain…

The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra.…

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a…

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M…

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial automorphism whose cycles all have the same length. In this paper we investigate the existence of semiregular…

In this note we prove: {\it Let $D$ be a 2-strong digraph of order $n$ such that its $n-1$ vertices have degrees at least $n+k$ and the remaining vertex $z$ has degree at least $n-k-4$, where $k$ is a positive integer. If $D$ contains a…

We give a geometric interpretation of the reciprocal complement of an integral domain $D$ in the case $D$ is a one-dimensional finitely generated algebra over an algebraically closed field.

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar…

We unconditionally determine $I_\Q(d)$, the set of possible prime degrees of cyclic $K$-isogneies of elliptic curves with $\Q$-rational $j$-invariants and without complex multiplication over number fields $K$ of degree $\leq d$, for $d\leq…

Motivated by the remarkable interplay between (chordal) graphs and matrix algebra, we associate to each graph a so-called completion number that might encode some aspects of that interplay. We show that this number is not trivial, and we…

In this paper we deal with a classical problem in elementary number theory, namely repeating decimals. We show how the digits of the period of the decimal representation of any fraction $\frac{k}{m}$, where $k$ and $m$ are positive integers…

Extending the work of Godsil and others, we investigate the notion of the inverse of a graph (specifically, of bipartite graphs with a unique perfect matching). We provide a concise necessary and sufficient condition for the invertibility…

In this paper we determine sufficient conditions for a quaternion algebra to split over a quadratic field. In the last section of the paper, we find a class of division symbol algebras of degree $n$ (where $n$ is a positive integer, $n\geq…

Let k a characteristic zero field. We give a characterization for the finite quiver k-algebras, based on double derivations. More precisely, we prove that if an associative and unitary k-algebra have a family of double derivations…