scholarly journals Infinite Graphs with Finite 2-Distinguishing Cost

10.37236/4263 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Debra Boutin ◽  
Wilfried Imrich
Keyword(s):  
Automorphism Group ◽  
Linear Growth ◽  
Infinite Graphs ◽  
Minimum Size ◽  
Locally Finite ◽  
Determining Set ◽  
The Cost

A graph $G$ is said to be 2-distinguishable if there is a labeling of the vertices with two labels such that only the trivial automorphism preserves the labels. Call the minimum size of a label class in such a labeling of $G$ the cost of 2-distinguishing $G$.We show that the connected, locally finite, infinite graphs with finite 2-distinguishing cost are exactly the graphs with countable automorphism group. Further we show that in such graphs the cost is less than three times the size of a smallest determining set. We also another, sharper bound on the 2-distinguishing cost, in particular for graphs of linear growth.

scholarly journals Bounding the Distinguishing Number of Infinite Graphs and Permutation Groups

10.37236/3046 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Simon M. Smith ◽  
Mark E. Watkins
Keyword(s):  
Connected Graph ◽  
Automorphism Groups ◽  
Cardinal Number ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Finite Radius ◽  
Locally Finite ◽  
Distinguishing Number ◽  
Identity Permutation ◽  
Imprimitive Group

A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is not $k$-distinguishable but in which every ball of finite radius is $k$-distinguishable.In the second half of this paper we show that a large distinguishing number for an imprimitive group $G$ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action by $G$ on the corresponding system of imprimitivity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.

scholarly journals Topological Infinite Gammoids, and a New Menger-Type Theorem for Infinite Graphs

10.37236/6083 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Johannes Carmesin
Keyword(s):  
Type Theorem ◽  
Main Tool ◽  
Infinite Graphs ◽  
Disjoint Paths ◽  
Natural Topology ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Vertex Sets ◽  
Special Case

Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid.As our main tool, we prove for any infinite graph $G$ with vertex-sets $A$ and $B$, if every finite subset of $A$ is linked to $B$ by disjoint paths, then the whole of $A$ can be linked to the closure of $B$ by disjoint paths or rays in a natural topology on $G$ and its ends.This latter theorem implies the topological Menger theorem of Diestel for locally finite graphs. It also implies a special case of the infinite Menger theorem of Aharoni and Berger.

scholarly journals Lower Bounds for Identifying Codes in some Infinite Grids

10.37236/394 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ryan Martin ◽  
Brendon Stanton
Keyword(s):  
Lower Bounds ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Locally Finite ◽  
Identifying Codes ◽  
Identifying Code ◽  
Hexagonal Grids ◽  
Closed Neighborhood ◽  
Definition Of ◽  
Infinite Grids

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

Derivations and Automorphism Group of Completed Witt Lie Algebra

2012 ◽  
Vol 19 (03) ◽  
pp. 581-590 ◽  
Author(s):  
Yongping Wu ◽  
Ying Xu ◽  
Lamei Yuan
Keyword(s):  
Finite Element ◽  
Automorphism Group ◽  
Lie Algebra ◽  
Cohomology Group ◽  
Locally Finite ◽  
Simple Lie Algebra ◽  
Derivation Algebra ◽  
First Cohomology Group

In this paper, a simple Lie algebra, referred to as the completed Witt Lie algebra, is introduced. Its derivation algebra and automorphism group are completely described. As a by-product, it is obtained that the first cohomology group of this Lie algebra with coefficients in its adjoint module is trivial. Furthermore, we completely determine the conjugate classes of this Lie algebra under its automorphism group, and also obtain that this Lie algebra does not contain any nonzero ad -locally finite element.

scholarly journals THE AUTOMORPHISM GROUP OF A SHIFT OF LINEAR GROWTH: BEYOND TRANSITIVITY

2015 ◽  
Vol 3 ◽  
Author(s):  
VAN CYR ◽  
BRYNA KRA
Keyword(s):  
Automorphism Group ◽  
Finite Groups ◽  
Linear Growth ◽  
Finite Alphabet ◽  
Finitely Generated ◽  
Complexity Function ◽  
Algebraic Properties

For a finite alphabet ${\mathcal{A}}$ and shift $X\subseteq {\mathcal{A}}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group $\text{Aut}(X)$. For such systems, we show that every finitely generated subgroup of $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if $X$ is transitive, then $\text{Aut}(X)$ is virtually $\mathbb{Z}$; if $X$ has dense aperiodic points, then $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$. We also classify all finite groups that arise as the automorphism group of a shift.

scholarly journals Discrete Schrödinger operators on a graph

1992 ◽  
Vol 125 ◽  
pp. 141-150 ◽  
Author(s):  
Polly Wee Sy ◽  
Toshikazu Sunada
Keyword(s):  
Riemannian Manifold ◽  
Automorphism Group ◽  
Connected Graph ◽  
Spectral Properties ◽  
Orbit Space ◽  
Finite Graph ◽  
Discrete Analogue ◽  
Discrete Laplacian ◽  
Locally Finite ◽  
Discrete Schrödinger Operators

In this paper, we study some spectral properties of the discrete Schrödinger operator = Δ + q defined on a locally finite connected graph with an automorphism group whose orbit space is a finite graph.The discrete Laplacian and its generalization have been explored from many different viewpoints (for instance, see [2] [4]). Our paper discusses the discrete analogue of the results on the bottom of the spectrum established by T. Kobayashi, K. Ono and T. Sunada [3] in the Riemannian-manifold-setting.

scholarly journals Symmetry Breaking in Tournaments

10.37236/3182 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Antoni Lozano
Keyword(s):  
Symmetry Breaking ◽  
Upper Bounds ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Nontrivial Automorphism ◽  
Determining Set ◽  
Strong Tournament

We provide upper bounds for the determining number and the metric dimension of tournaments. A set of vertices $S \subseteq V(T)$ is a determining set for a tournament $T$ if every nontrivial automorphism of $T$ moves at least one vertex of $S$, while $S$ is a resolving set for $T$ if every two distinct vertices in $T$ have different distances to some vertex in $S$. We show that the minimum size of a determining set for an order $n$ tournament (its determining number) is bounded by $\lfloor n/3 \rfloor$, while the minimum size of a resolving set for an order $n$ strong tournament (its metric dimension) is bounded by $\lfloor n/2 \rfloor$. Both bounds are optimal.

scholarly journals Infinite Graphical Frobenius Representations

10.37236/7294 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Mark E. Watkins
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Polynomial Growth ◽  
Free Products ◽  
Locally Finite ◽  
Finitely Generated Groups ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Planar Maps ◽  
Transitive Graphs

A graphical Frobenius representation (GFR) of a Frobenius (permutation) group $G$ is a graph $\Gamma$ whose automorphism group Aut$(\Gamma)$ acts as a Frobenius permutation group on the vertex set of $\Gamma$, that is, Aut$(\Gamma)$ acts vertex-transitively with the property that all nonidentity automorphisms fix either exactly one or zero vertices and there are some of each kind. The set $K$ of all fixed-point-free automorphisms together with the identity is called the kernel of $G$. Whenever $G$ is finite, $K$ is a regular normal subgroup of $G$ (F. G. Frobenius, 1901), in which case $\Gamma$ is a Cayley graph of $K$. The same holds true for all the infinite instances presented here.Infinite, locally finite, vertex-transitive graphs can be classified with respect to (i) the cardinality of their set of ends and (ii) their growth rate. We construct families of infinite GFRs for all possible combinations of these two properties. There exist infinitely many GFRs with polynomial growth of degree $d$ for every positive integer $d$, and there exist infinite families of GFRs of exponential growth, both $1$-ended and infinitely-ended, that underlie infinite chiral planar maps. There also exist GFRs of free products of finitely many finitely generated groups. Graphs of connectivity 1 having a Frobenius automorphism group are characterized.

scholarly journals COHERENT EXTENSION OF PARTIAL AUTOMORPHISMS, FREE AMALGAMATION AND AUTOMORPHISM GROUPS

2019 ◽  
Vol 85 (1) ◽  
pp. 199-223 ◽  
Author(s):  
DAOUD SINIORA ◽  
SŁAWOMIR SOLECKI
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Extension Property ◽  
Finite Subgroup ◽  
Homogeneous Structure ◽  
Extension Theorems ◽  
Locally Finite ◽  
Finite Structures ◽  
Image Position ◽  
Fraïssé Class

AbstractWe give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all ${\cal F}$-free structures (in the Herwig–Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

M-Series with Adaptive Patterning for High-Yield Fan-Out SIP

2016 ◽  
Vol 2016 (DPC) ◽  
pp. 000751-000773
Author(s):  
Craig Bishop ◽  
Suresh Jayaraman ◽  
Boyd Rogers ◽  
Chris Scanlan ◽  
Tim Olson
Keyword(s):  
New Technology ◽  
Minimum Size ◽  
High Yield ◽  
Wafer Level ◽  
Design Rules ◽  
Optical Scanner ◽  
Wafer Level Packaging ◽  
And Performance ◽  
The Cost ◽  
Semiconductor Chips

Fan-Out Wafer Level Packaging (FOWLP) holds immediate promise for packaging semiconductor chips with higher interconnect density than the incumbent Wafer Level Chip Scale Packaging (WLCSP). FOWLP enables size and performance capabilities similar to WLCSP, while extending capabilities to include multi-device system-in-packages. FOWLP can support applications that integrate multiple heterogeneously processed die at lower cost than 2.5D silicon interposer technologies. Current industry challenges with die position yield after die placement and molding result in low-density design rules and the high-cost of accurate die placement. Efficiently handling die shift is essential for making FOWLP cost-competitive with other technologies such as FCCSP and QFN. This presentation will provide an overview of Adaptive Patterning, a new technology for overcoming variability of die positions after placement and molding. In this process, an optical scanner is used to measure the true XY position and rotation of each die after panelization. The die measurements are then fed into a proprietary software engine that generates a unique pattern for each package. The resulting patterns are dispatched to a lithography system, which dynamically implements the unique patterns for all packages within a panel. For system-in-packages, this process offers a unique advantage over a fixed pattern: each die shift can be handled independently. With a fixed pattern, the design tolerances need to be large enough for all die to shift in opposing directions, otherwise yield loss in incurred. With Adaptive Patterning, vias and RDL features remain at minimum size and are matched to the measured die shift. The die-to-die interconnects are dynamically generated and account for the unique position of each die. Thus, Adaptive Patterning retains the same high-density design rules regardless of how many die are in a package. Adaptive Patterning provides the capability to use high-throughput die placement to drive down cost, while enabling higher-density system-in-package interconnect. With this technology the industry can finally realize the cost, flexibility, and form factor benefits of FOWLP.

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