scholarly journals Symmetry Breaking in Graphs

10.37236/1242 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Michael O. Albertson ◽  
Karen L. Collins
Keyword(s):  
Symmetry Breaking ◽  
Complete Graph ◽  
Subject Classification ◽  
Distinguishing Number

A labeling of the vertices of a graph G, $\phi :V(G) \rightarrow \{1,\ldots,r\}$, is said to be $r$-distinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by $D(G)$, is the minimum $r$ such that $G$ has an $r$-distinguishing labeling. The distinguishing number of the complete graph on $t$ vertices is $t$. In contrast, we prove (i) given any group $\Gamma$, there is a graph $G$ such that $Aut(G) \cong \Gamma$ and $D(G)= 2$; (ii) $D(G) = O(log(|Aut(G)|))$; (iii) if $Aut(G)$ is abelian, then $D(G) \leq 2$; (iv) if $Aut(G)$ is dihedral, then $D(G) \leq 3$; and (v) If $Aut(G) \cong S_4$, then either $D(G) = 2$ or $D(G) = 4$. Mathematics Subject Classification 05C,20B,20F,68R

scholarly journals Graph connectivity and Wiener index

2006 ◽  
Vol 133 (31) ◽  
pp. 1-5 ◽  
Author(s):  
I. Gutman ◽  
S. Zhang
Keyword(s):  
Complete Graph ◽  
Wiener Index ◽  
Graph Connectivity ◽  
Subject Classification ◽  
Edge Connectivity

The graphs with a given number n of vertices and given (vertex or edge) connectivity k, having minimum Wiener index are determined. In both cases this is Kk + (K1 U Kn-k-1), the graph obtained by connecting all vertices of the complete graph Kk with all vertices of the graph whose two components are Kn-k-1 and K1. AMS Mathematics Subject Classification (2000): 05C12, 05C40 05C35.

scholarly journals Symmetry breaking in planar and maximal outerplanar graphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950008
Author(s):  
Saeid Alikhani ◽  
Samaneh Soltani
Keyword(s):  
Symmetry Breaking ◽  
Index Formula ◽  
Outerplanar Graphs ◽  
Distinguishing Number ◽  
Maximal Outerplanar Graphs ◽  
Edge Labeling

The distinguishing number (index) [Formula: see text] ([Formula: see text]) of a graph [Formula: see text] is the least integer [Formula: see text] such that [Formula: see text] has a vertex (edge) labeling with [Formula: see text] labels that is preserved only by a trivial automorphism. In this paper, we consider the maximal outerplanar graphs (MOP graphs) and show that MOP graphs, except [Formula: see text], can be distinguished by at most two vertex (edge) labels. We also compute the distinguishing number and the distinguishing index of Halin and Mycielskian graphs.

scholarly journals Quantum Spins and Random Loops on the Complete Graph

2019 ◽  
Vol 375 (3) ◽  
pp. 1629-1663
Author(s):  
Jakob E. Björnberg ◽  
Jürg Fröhlich ◽  
Daniel Ueltschi
Keyword(s):  
Phase Transitions ◽  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Spin Density ◽  
Generating Function ◽  
Complete Graph ◽  
Critical Exponents ◽  
Systematic Analysis ◽  
Dirichlet Distributions ◽  
Quantum Spins

AbstractWe present a systematic analysis of quantum Heisenberg-, xy- and interchange models on the complete graph. These models exhibit phase transitions accompanied by spontaneous symmetry breaking, which we study by calculating the generating function of expectations of powers of the averaged spin density. Various critical exponents are determined. Certain objects of the associated loop models are shown to have properties of Poisson–Dirichlet distributions.

COMMON MULTIPLES OF COMPLETE GRAPHS

2003 ◽  
Vol 86 (2) ◽  
pp. 302-326 ◽  
Author(s):  
DARRYN BRYANT ◽  
BARBARA MAENHAUT
Keyword(s):  
Complete Graph ◽  
Subject Classification ◽  
Complete Graphs ◽  
Steiner Triple Systems ◽  
Common Multiple ◽  
Triple Systems ◽  
Johnson Graphs ◽  
Positive Integers

A graph $H$ is said to divide a graph $G$ if there exists a set $S$ of subgraphs of $G$, all isomorphic to $H$, such that the edge set of $G$ is partitioned by the edge sets of the subgraphs in $S$. Thus, a graph $G$ is a common multiple of two graphs if each of the two graphs divides $G$. This paper considers common multiples of a complete graph of order $m$ and a complete graph of order $n$. The complete graph of order $n$ is denoted $K_n$. In particular, for all positive integers $n$, the set of integers $q$ for which there exists a common multiple of $K_3$ and $K_n$ having precisely $q$ edges is determined.It is shown that there exists a common multiple of $K_3$ and $K_n$ having $q$ edges if and only if $q \equiv 0 \, ({\rm mod}\, 3)$, $q \equiv 0 \, ({\rm mod}\, \binom n2)$ and(1) $q \neq 3 \binom n2$ when $n \equiv 5 \, ({\rm mod}\, 6)$; (2) $q \geq (n + 1) \binom n2$ when $n$ is even; (3) $q \notin \{36, 42, 48\}$ when $n = 4$.The proof of this result uses a variety of techniques including the use of Johnson graphs, Skolem and Langford sequences, and equitable partial Steiner triple systems.2000 Mathematical Subject Classification: 05C70, 05B30, 05B07.

Symmetry breaking in convergent-beam diffraction

1989 ◽  
Vol 47 ◽  
pp. 480-481
Author(s):  
D.J. Eaglesham
Keyword(s):  
Symmetry Breaking ◽  
Point Group ◽  
X Ray Diffraction ◽  
Multiple Diffraction ◽  
Space Groups ◽  
Atomic Displacements ◽  
Small Probe ◽  
Phase Sensitive ◽  
Convergent Beam

Convergent Beam Electron Diffraction is now almost routinely used in the determination of the point- and space-groups of crystalline samples. In addition to its small-probe capability, CBED is also postulated to be more sensitive than X-ray diffraction in determining crystal symmetries. Multiple diffraction is phase-sensitive, so that the distinction between centro- and non-centro-symmetric space groups should be trivial in CBED: in addition, the stronger scattering of electrons may give a general increase in sensitivity to small atomic displacements. However, the sensitivity of CBED symmetry to the crystal point group has rarely been quantified, and CBED is also subject to symmetry-breaking due to local strains and inhomogeneities. The purpose of this paper is to classify the various types of symmetry-breaking, present calculations of the sensitivity, and illustrate symmetry-breaking by surface strains.CBED symmetry determinations usually proceed by determining the diffraction group along various zone axes, and hence finding the point group. The diffraction group can be found using either the intensity distribution in the discs

Centrosome Aurora A gradient ensures single polarity axis in C. elegans embryos

2020 ◽  
Vol 48 (3) ◽  
pp. 1243-1253 ◽  
Author(s):  
Sukriti Kapoor ◽  
Sachin Kotak
Keyword(s):  
Symmetry Breaking ◽  
Cell Fate ◽  
Cell Cortex ◽  
Axis Formation ◽  
Aurora A Kinase ◽  
Aurora A ◽  
C Elegans ◽  
Cell Embryo ◽  
Polarity Establishment

Cellular asymmetries are vital for generating cell fate diversity during development and in stem cells. In the newly fertilized Caenorhabditis elegans embryo, centrosomes are responsible for polarity establishment, i.e. anterior–posterior body axis formation. The signal for polarity originates from the centrosomes and is transmitted to the cell cortex, where it disassembles the actomyosin network. This event leads to symmetry breaking and the establishment of distinct domains of evolutionarily conserved PAR proteins. However, the identity of an essential component that localizes to the centrosomes and promotes symmetry breaking was unknown. Recent work has uncovered that the loss of Aurora A kinase (AIR-1 in C. elegans and hereafter referred to as Aurora A) in the one-cell embryo disrupts stereotypical actomyosin-based cortical flows that occur at the time of polarity establishment. This misregulation of actomyosin flow dynamics results in the occurrence of two polarity axes. Notably, the role of Aurora A in ensuring a single polarity axis is independent of its well-established function in centrosome maturation. The mechanism by which Aurora A directs symmetry breaking is likely through direct regulation of Rho-dependent contractility. In this mini-review, we will discuss the unconventional role of Aurora A kinase in polarity establishment in C. elegans embryos and propose a refined model of centrosome-dependent symmetry breaking.

A Symmetry-Breaking Theory of False Pop Out

2011 ◽  
Author(s):  
Kimberley D. Orsten ◽  
Mary C. Portillo ◽  
James R. Pomerantz
Keyword(s):  
Symmetry Breaking
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