scholarly journals Distinguishing Graphs of Maximum Valence 3

10.37236/7281 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Svenja Hüning ◽  
Wilfried Imrich ◽  
Judith Kloas ◽  
Hannah Schreber ◽  
Thomas W. Tucker
Keyword(s):  
Complete Classification ◽  
Connected Graphs ◽  
Distinguishing Number ◽  
Delta G

The distinguishing number $D(G)$ of a graph $G$ is the smallest number of colors that is needed to color the vertices such that the only color-preserving automorphism fixes all vertices. We give a complete classification for all connected graphs $G$ of maximum valence $\Delta(G) = 3$ and distinguishing number $D(G) = 3$. As one of the consequences we show that all infinite connected graphs with $\Delta(G) = 3$ are $2$-distinguishable.

scholarly journals The Distinguishing Chromatic Number

10.37236/1042 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Karen L. Collins ◽  
Ann N. Trenk
Keyword(s):  
Chromatic Number ◽  
Connected Graphs ◽  
Distinguishing Number

In this paper we define and study the distinguishing chromatic number, $\chi_D(G)$, of a graph $G$, building on the work of Albertson and Collins who studied the distinguishing number. We find $\chi_D(G)$ for various families of graphs and characterize those graphs with $\chi_D(G)$ $ = |V(G)|$, and those trees with the maximum chromatic distingushing number for trees. We prove analogs of Brooks' Theorem for both the distinguishing number and the distinguishing chromatic number, and for both trees and connected graphs. We conclude with some conjectures.

scholarly journals Distinguishing Maps II: General Case

10.37236/3410 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Thomas W. Tucker
Keyword(s):  
Automorphism Group ◽  
Complete Classification ◽  
Distinguishing Number ◽  
Nonidentity Element ◽  
Vertex Set ◽  
Group A

A group $A$ acting faithfully on a set $X$ has  distinguishing number $k$, written $D(A,X)=k$, if there is a coloring of the elements of $X$ with $k$ colors such that no nonidentity element of $A$ is color-preserving, and no such coloring with fewer than $k$ colors exists.  Given a map $M$ with vertex set $V$ and automorphism group $Aut(M)$, let $D(M)=D(Aut(M),V)$. If $M$ is orientable, let $D^+(M)=D(Aut^+(M),V)$, where $Aut^+(M)$ is the group of orientation-preserving automorphisms.   In a previous paper, the author showed there are four maps $M$ with $D^+(M)>2$.  In this paper,  a complete classification is given for the graphs underlying maps with $D(M)>2$. There are $31$ such graphs, $22$ having no vertices of valence $1$ or $2$, and all have at most $10$ vertices.

scholarly journals Bounds for Distinguishing Invariants of Infinite Graphs

10.37236/6362 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Wilfried Imrich ◽  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Mohammad Hadi Shekarriz
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Infinite Graphs ◽  
Chromatic Index ◽  
Distinguishing Number ◽  
Minimum Number ◽  
Edge Colourings ◽  
Delta G ◽  
Vertex Colouring

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $\chi_D(G)$, and the distinguishing chromatic index $\chi'_D(G)$, for vertex and edge colourings, respectively. We show that $\chi_D(G)\leq 2\Delta(G)-1$ for graphs with a finite maximum degree $\Delta(G)$, and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that $\chi'_D(G)\leq \chi'(G)+1$, where $\chi'(G)$ is the chromatic index of $G$, and we prove a similar result $\chi''_D(G)\leq \chi''(G)+1$ for proper total colourings. A number of conjectures are formulated.

scholarly journals On Asymmetric Colourings of Claw-Free Graphs

10.37236/8886 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Wilfried Imrich ◽  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Mariusz Woźniak
Keyword(s):  
Maximum Degree ◽  
Distinguishing Number ◽  
Identity Automorphism ◽  
Minimum Number ◽  
Large Motion ◽  
Delta G ◽  
Vertex Colouring

A vertex colouring of a graph is asymmetric if it is preserved only by the identity automorphism. The minimum number of colours needed for an asymmetric colouring of a graph $G$ is called the asymmetric colouring number or distinguishing number $D(G)$ of $G$. It is well known that $D(G)$ is closely related to the least number of vertices moved by any non-identity automorphism, the so-called motion $m(G)$ of $G$. Large motion is usually correlated with small $D(G)$. Recently, Babai posed the question whether there exists a function $f(d)$ such that every connected, countable graph $G$ with maximum degree $\Delta(G)\leq d$ and motion $m(G)>f(d)$ has an asymmetric $2$-colouring, with at most finitely many exceptions for every degree. We prove the following result: if $G$ is a connected, countable graph of maximum degree at most 4, without an induced claw $K_{1,3}$, then $D(G)= 2$ whenever $m(G)>2$, with three exceptional small graphs. This answers the question of Babai for $d=4$ in the class of~claw-free graphs.

scholarly journals Trees with an On-Line Degree Ramsey Number of Four

10.37236/660 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
David Rolnick
Keyword(s):  
Maximum Degree ◽  
Ramsey Theory ◽  
Complete Classification ◽  
Ramsey Number ◽  
On Line ◽  
Delta G ◽  
Goal Graph ◽  
Monochromatic Copy

On-line Ramsey theory studies a graph-building game between two players. The player called Builder builds edges one at a time, and the player called Painter paints each new edge red or blue after it is built. The graph constructed is called the background graph. Builder's goal is to cause the background graph to contain a monochromatic copy of a given goal graph, and Painter's goal is to prevent this. In the $S_k$-game variant of the typical game, the background graph is constrained to have maximum degree no greater than $k$. The on-line degree Ramsey number $\mathring{R}_{\Delta}(G)$ of a graph $G$ is the minimum $k$ such that Builder wins an $S_k$-game in which $G$ is the goal graph. Butterfield et al. previously determined all graphs $G$ satisfying $\mathring{R}_{\Delta}(G)\le 3$. We provide a complete classification of trees $T$ satisfying $\mathring{R}_{\Delta}(T)=4$.

scholarly journals The Distinguishing Index of Infinite Graphs

10.37236/3933 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Izak Broere ◽  
Monika Pilśniak
Keyword(s):  
Random Graph ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Distinguishing Number ◽  
General Bound ◽  
Delta G ◽  
Edge Colouring

The  distinguishing index $D^\prime(G)$ of a graph $G$ is the least cardinal $d$ such that $G$ has an edge colouring with $d$ colours that is only preserved by the trivial automorphism. This is similar to the notion of the distinguishing number $D(G)$ of a graph $G$, which is defined with respect to vertex colourings.We derive several bounds for infinite graphs, in particular, we prove the general bound $D^\prime(G)\leq\Delta(G)$ for an arbitrary infinite graph. Nonetheless,  the distinguishing index is at most two for many countable graphs, also for the infinite random graph and for uncountable tree-like graphs.We also investigate the concept of the motion of edges and its relationship with the Infinite Motion Lemma. 

Quantitative Ranking of Ligand Binding Kinetics with a Multiscale Milestoning Simulation Approach

2018 ◽  
Author(s):  
Benjamin R. Jagger ◽  
Christoper T. Lee ◽  
Rommie Amaro
Keyword(s):  
Molecular Dynamics ◽  
Drug Discovery ◽  
Small Molecule ◽  
Brownian Dynamics ◽  
Model Simulation ◽  
Computational Cost ◽  
Lead Selection ◽  
Delta G ◽  
Dynamics Simulations

<p>The ranking of small molecule binders by their kinetic (kon and koff) and thermodynamic (delta G) properties can be a valuable metric for lead selection and optimization in a drug discovery campaign, as these quantities are often indicators of in vivo efficacy. Efficient and accurate predictions of these quantities can aid the in drug discovery effort, acting as a screening step. We have previously described a hybrid molecular dynamics, Brownian dynamics, and milestoning model, Simulation Enabled Estimation of Kinetic Rates (SEEKR), that can predict kon’s, koff’s, and G’s. Here we demonstrate the effectiveness of this approach for ranking a series of seven small molecule compounds for the model system, -cyclodextrin, based on predicted kon’s and koff’s. We compare our results using SEEKR to experimentally determined rates as well as rates calculated using long-timescale molecular dynamics simulations and show that SEEKR can effectively rank the compounds by koff and G with reduced computational cost. We also provide a discussion of convergence properties and sensitivities of calculations with SEEKR to establish “best practices” for its future use.</p>

scholarly journals The Formula to Count The Number of Vertices Labeled Order Six Connected Graphs with Maximum Thirty Edges without Loops

2021 ◽  
Vol 1751 ◽  
pp. 012023
Author(s):  
F C Puri ◽  
Wamiliana ◽  
M Usman ◽  
Amanto ◽  
M Ansori ◽  
...  
Keyword(s):  
Connected Graphs

scholarly journals Psychometric analysis of the Brazilian-version Kidscreen-27 questionnaire

2021 ◽  
Vol 19 (1) ◽  
Author(s):  
Pablo Magno da Silveira ◽  
Alexsandra da Silva Bandeira ◽  
Marcus Vinicius Veber Lopes ◽  
Adriano Ferreti Borgatto ◽  
Kelly Samara da Silva
Keyword(s):  
Public Schools ◽  
Construct Validity ◽  
Internal Consistency ◽  
Intraclass Correlation ◽  
Well Being ◽  
Discriminatory Power ◽  
Brazilian Version ◽  
Related Quality ◽  
Health Related ◽  
Delta G

Abstract Background The objective of this study was to verify the reliability, discriminatory power and construct validity of the Kidscreen-27 questionnaire in Brazilian adolescents. Methods Adolescents that participated of the pilot study (210 adolescents; 52.9% boys; 13.7 years old) and of the baseline (816 participants; 52.7% girls; 13.1 years old) of the Movimente Project in 2016/2017 composed the sample of the present study. This project was carried out in six public schools in the city of Florianópolis, Santa Catarina, Brazil. Test–retest reproducibility was assessed by the intraclass correlation coefficient and Gwet coefficient; internal consistency through McDonald's Omega; Hankins' Delta G coefficient verified the scale's discriminatory power and; confirmatory factor analysis to assess construct validity. Results Reproducibility values ranged from 0.71 to 0.78 for the dimensions (ICC), and ranged from 0.60 to 0.83 for the items (Gwet). McDonald's Ômega (0.82–0.91) for internal consistency measures. Discriminatory power ranging from 0.94 for the dimension Social Support and Friends to 0.98 for Psychological Well-Being. The factorial loads were > 0.40, except for item 19 (0.36). The fit quality indicators of the model were adequate (X2[df] = 1022.89 [311], p < 0.001; RMSEA = 0.053 (0.049–0.087); CFI = 0.988; TLI = 0.987), confirming the five-factor structure originally proposed. Conclusions The Brazilian-version Kidscreen-27 achieved good levels of reproducibility, internal consistency, discriminatory power and construct validity. Its use is adequate to measure the health-related quality of life of adolescents in the Brazilian context.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

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