Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0504151

The Galaxies of Nonstandard Enlargements of Infinite and Transfinite grpahs of Higher Ranks: II

A. H. Zemanian
View original source ↗Read source document ↗Export metadata ↓

Abstract

This report is an improvement of a prior report (Report 814). It sharpens the principal theorem (Theorem 5.1 of Report 814) and also simplifies its proof. There are also several minor changes involving clarifications and corrections of misprints. The prior abstract remins the same as follows: In a previous work, the galaxies of the nonstandard enlargements of connected, conventionally infinite and also of transfinite graphs of the first rank of transfiniteness were defined, examined, and illustrated by some examples. In this work it is shown how the results of the prior work extend to transfinite graphs of higher ranks. Among those results are following: Any such enlargement either has exactly one galaxy, its principal one, ot it has infinitely many such galaxies. In the latter case, the galaxies are partially ordered by their "closeness" to the principal galaxy. Also, certain sequences of galaxies whose members are totally ordered by that "closeness" property are identified.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

A. H. Zemanian. 2005-04-07. The Galaxies of Nonstandard Enlargements of Infinite and Transfinite grpahs of Higher Ranks: II. https://arxiv.org/abs/math/0504151

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Small doublings in abelian groups of prime power torsion

Let $A$ be a subset of $G$, where $G$ is a finite abelian group of torsion $r$. It was conjectured by Ruzsa that if $|A+A|\leq K|A|$, then $A$ is contained in a coset of $G$ of size at most $r^{CK}|A|$ for some constant $C$. The case $r=2$ received considerable attention in a sequence of papers, and was resolved by Green and Tao. Recently, Even-Zohar and Lovett settled the case when $r$ is a prime. In this paper, we confirm the conjecture when $r$ is a power of prime. In particular, the bound we obtain is tight.

math.CO↗
arXiv2026

Graph Polynomial for Colored Embedded Graphs: A Topological Approach

We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in physics. We also analyze a variant of these polynomials for colored embedded graphs. This is used to describe the change in the polynomial under basic graph theoretic operations. We conclude with several applications of this polynomial including detection of certain classes of graphs and the connection of this polynomial with topological entanglement entropy.

math.CO↗
arXiv2026

Schur positivity of the spiders $S(a,2,1)$ and $S(a,4,1)$ via noncommutative symmetric functions

We prove that the spider graphs $S(a,2,1)$ and $S(a,4,1)$ are Schur positive for all integers $a\ge1$. Together with the known $e$-positivity results, this completes the $e$- and Schur-positivity classification of both families. Our approach uses noncommutative symmetric functions, including a particularly simple ribbon expansion for the path lift with coefficients given by powers of two. We give a new proof of the Shareshian--Wachs path formula at $t=1$ and construct corresponding lifts for spiders. The Littlewood--Richardson rule converts their ribbon expansions into a general Schur-coefficient formula in terms of weighted Yamanouchi words. Mass-preserving multi-injections and a reduction to finitely many inequalities in degree $10$ then prove the required positivity; exact computer verification of these inequalities completes the proof in full generality.

math.CO↗