Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userUbiquity in graphs I: Topological ubiquity of trees
126
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $alpha G$ is the disjoint union of $alpha$ many copies of $G$. The emph{Ubiquity Conjecture} of Andreae, a well-known open problem in the theory of infinite graphs, asserts that every locally finite connected graph is ubiquitous with respect to the minor relation. In this paper, which is the first of a series of papers making progress towards the Ubiquity Conjecture, we show that all trees are ubiquitous with respect to the topological minor relation, irrespective of their cardinality. This answers a question of Andreae from 1979.
rate research
Read More
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 disjoint union of $alpha$ many copies of $G$. A well-known conjecture of Andreae is that every locally finite connected graph is $preceq$-ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph~$G$ which implies that $G$ is $preceq$-ubiquitous. In particular this implies that the full grid is $preceq$-ubiquitous.
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 graph is ubiquitous. In this paper we show that locally finite graphs admitting a certain type of tree-decomposition, which we call an extensive tree-decomposition, are ubiquitous. In particular this includes all locally finite graphs of finite tree-width, and also all locally finite graphs with finitely many ends, all of which have finite degree. It remains an open question whether every locally finite graph admits an extensive tree-decomposition.
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.
Non-topological gauged soliton solutions called Q-balls arise in many scalar field theories that are invariant under a U(1) gauge symmetry. The related, but qualitatively distinct, Q-shell solitons have only been shown to exist for special potentials. We investigate gauged solitons in a generic sixth-order polynomial potential (that contains the leading effects of many effective field theories) and show that this potential generically allows for both Q-balls and Q-shells. We argue that Q-shell solutions occur in many, and perhaps all, potentials that have previously only been shown to contain Q-balls. We give simple analytic characterizations of these Q-shell solutions, leading to excellent predictions of their physical properties.
We present the results of a study of a statistically significant sample of galaxies which clearly demonstrate that supermassive black holes are generically present in all morphological types. Our analysis is based on the quantitative morphological classification of 1.12 million galaxies in the SDSS DR7 and on the detection of black hole activity via two different methods, the first one based on their X-ray/radio emission and the second one based on their mid-infrared colors. The results of the first analysis confirm the correlation between black hole and total stellar mass for 8 galaxies and includes one galaxy classified as bulgeless. The results of our second analysis, consisting of 15,991 galaxies, show that galaxies hosting a supermassive black hole follow the same morphological distribution as the general population of galaxies in the same redshift range. In particular, the fraction of bulgeless galaxies, 1,450 galaxies or 9 percent, is found to be the same as in the general population. We also present the correlation between black hole and total stellar mass for 6,247 of these galaxies. Importantly, whereas previous studies were limited to primarily bulge-dominated systems, our study confirms this relationship to all morphological types, in particular, to 530 bulgeless galaxies. Our results indicate that the true correlation that exists for supermassive black holes and their host galaxies is between the black hole mass and the total stellar mass of the galaxy and hence, we conclude that the previous assumption that the black hole mass is correlated with the bulge mass is only approximately correct.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
