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\documentclass[a4paper,11pt]{report} \usepackage{a4wide} \newcommand{\bbZ}{\mathbb{Z}} \usepackage[top=37mm,bottom=37mm,left=27mm,right=27mm]{geometry} \sloppy \pagestyle{myheadings} \usepackage{amssymb} \usepackage[utf8]{inputenc} \usepackage{makeidx} \makeindex \usepackage{color} \definecolor{FireBrick}{rgb}{0.5812,0.0074,0.0083} \definecolor{RoyalBlue}{rgb}{0.0236,0.0894,0.6179} \definecolor{RoyalGreen}{rgb}{0.0236,0.6179,0.0894} \definecolor{RoyalRed}{rgb}{0.6179,0.0236,0.0894} \definecolor{LightBlue}{rgb}{0.8544,0.9511,1.0000} \definecolor{Black}{rgb}{0.0,0.0,0.0} \definecolor{linkColor}{rgb}{0.0,0.0,0.554} \definecolor{citeColor}{rgb}{0.0,0.0,0.554} \definecolor{fileColor}{rgb}{0.0,0.0,0.554} \definecolor{urlColor}{rgb}{0.0,0.0,0.554} \definecolor{promptColor}{rgb}{0.0,0.0,0.589} \definecolor{brkpromptColor}{rgb}{0.589,0.0,0.0} \definecolor{gapinputColor}{rgb}{0.589,0.0,0.0} \definecolor{gapoutputColor}{rgb}{0.0,0.0,0.0} %% for a long time these were red and blue by default, %% now black, but keep variables to overwrite \definecolor{FuncColor}{rgb}{0.0,0.0,0.0} %% strange name because of pdflatex bug: \definecolor{Chapter }{rgb}{0.0,0.0,0.0} \definecolor{DarkOlive}{rgb}{0.1047,0.2412,0.0064} \usepackage{fancyvrb} \usepackage{mathptmx,helvet} \usepackage[T1]{fontenc} \usepackage{textcomp} \usepackage[ pdftex=true, bookmarks=true, a4paper=true, pdftitle={Written with GAPDoc}, pdfcreator={LaTeX with hyperref package / GAPDoc}, colorlinks=true, backref=page, breaklinks=true, linkcolor=linkColor, citecolor=citeColor, filecolor=fileColor, urlcolor=urlColor, pdfpagemode={UseNone}, ]{hyperref} \newcommand{\maintitlesize}{\fontsize{50}{55}\selectfont} % write page numbers to a .pnr log file for online help \newwrite\pagenrlog \immediate\openout\pagenrlog =\jobname.pnr \immediate\write\pagenrlog{PAGENRS := [} \newcommand{\logpage}[1]{\protect\write\pagenrlog{#1, \thepage,}} %% were never documented, give conflicts with some additional packages \newcommand{\GAP}{\textsf{GAP}} %% nicer description environments, allows long labels \usepackage{enumitem} \setdescription{style=nextline} %% depth of toc \setcounter{tocdepth}{1} %% command for ColorPrompt style examples \newcommand{\gapprompt}[1]{\color{promptColor}{\bfseries #1}} \newcommand{\gapbrkprompt}[1]{\color{brkpromptColor}{\bfseries #1}} \newcommand{\gapinput}[1]{\color{gapinputColor}{#1}} \begin{document} \logpage{[ 0, 0, 0 ]} \begin{titlepage} \mbox{}\vfill \begin{center}{\maintitlesize \textbf{ Digraphs \mbox{}}}\\ \vfill \hypersetup{pdftitle= Digraphs } \markright{\scriptsize \mbox{}\hfill Digraphs \hfill\mbox{}} {\Huge \textbf{ Graphs, digraphs, and multidigraphs in \textsf{GAP} \mbox{}}}\\ \vfill {\Huge 1.13.1 \mbox{}}\\[1cm] { 27 September 2025 \mbox{}}\\[1cm] \mbox{}\\[2cm] {\Large \textbf{ Jan De Beule\\ \mbox{}}}\\ {\Large \textbf{ Julius Jonusas\\ \mbox{}}}\\ {\Large \textbf{ James Mitchell\\ \mbox{}}}\\ {\Large \textbf{ Wilf A. Wilson\\ \mbox{}}}\\ {\Large \textbf{ Michael Young\\ \mbox{}}}\\ {\Large \textbf{ Marina Anagnostopoulou\texttt{\symbol{45}}Merkouri\\ \mbox{}}}\\ {\Large \textbf{ Finn Buck\\ \mbox{}}}\\ {\Large \textbf{ Stuart Burrell\\ \mbox{}}}\\ {\Large \textbf{ Graham Campbell\\ \mbox{}}}\\ {\Large \textbf{ Raiyan Chowdhury\\ \mbox{}}}\\ {\Large \textbf{ Reinis Cirpons\\ \mbox{}}}\\ {\Large \textbf{ Ashley Clayton\\ \mbox{}}}\\ {\Large \textbf{ Tom Conti\texttt{\symbol{45}}Leslie\\ \mbox{}}}\\ {\Large \textbf{ Joseph Edwards\\ \mbox{}}}\\ {\Large \textbf{ Luna Elliott\\ \mbox{}}}\\ {\Large \textbf{ Jan Engelhardt\\ \mbox{}}}\\ {\Large \textbf{ Isuru Fernando\\ \mbox{}}}\\ {\Large \textbf{ Ewan Gilligan\\ \mbox{}}}\\ {\Large \textbf{ Gillis Frankie\\ \mbox{}}}\\ {\Large \textbf{ Sebastian Gutsche\\ \mbox{}}}\\ {\Large \textbf{ Samantha Harper\\ \mbox{}}}\\ {\Large \textbf{ Max Horn\\ \mbox{}}}\\ {\Large \textbf{ Harry Jack\\ \mbox{}}}\\ {\Large \textbf{ Christopher Jefferson\\ \mbox{}}}\\ {\Large \textbf{ Malachi Johns\\ \mbox{}}}\\ {\Large \textbf{ Olexandr Konovalov\\ \mbox{}}}\\ {\Large \textbf{ Hyeokjun Kwon\\ \mbox{}}}\\ {\Large \textbf{ Aidan Lau\\ \mbox{}}}\\ {\Large \textbf{ Andrea Lee\\ \mbox{}}}\\ {\Large \textbf{ Saffron McIver\\ \mbox{}}}\\ {\Large \textbf{ Seyyed Ali Mohammadiyeh\\ \mbox{}}}\\ {\Large \textbf{ Michael Orlitzky\\ \mbox{}}}\\ {\Large \textbf{ Matthew Pancer\\ \mbox{}}}\\ {\Large \textbf{ Markus Pfeiffer\\ \mbox{}}}\\ {\Large \textbf{ Daniel Pointon\\ \mbox{}}}\\ {\Large \textbf{ Pramoth Ragavan\\ \mbox{}}}\\ {\Large \textbf{ Lea Racine\\ \mbox{}}}\\ {\Large \textbf{ Christopher Russell\\ \mbox{}}}\\ {\Large \textbf{ Artur Schaefer\\ \mbox{}}}\\ {\Large \textbf{ Isabella Scott\\ \mbox{}}}\\ {\Large \textbf{ Kamran Sharma\\ \mbox{}}}\\ {\Large \textbf{ Finn Smith\\ \mbox{}}}\\ {\Large \textbf{ Ben Spiers\\ \mbox{}}}\\ {\Large \textbf{ Maria Tsalakou\\ \mbox{}}}\\ {\Large \textbf{ Agastyaa Vishvanath\\ \mbox{}}}\\ {\Large \textbf{ Meike Weiss\\ \mbox{}}}\\ {\Large \textbf{ Murray Whyte\\ \mbox{}}}\\ {\Large \textbf{ Fabian Zickgraf\\ \mbox{}}}\\ \hypersetup{pdfauthor= Jan De Beule\\ ; Julius Jonusas\\ ; James Mitchell\\ ; Wilf A. Wilson\\ ; Michael Young\\ ; Marina Anagnostopoulou\texttt{\symbol{45}}Merkouri\\ ; Finn Buck\\ ; Stuart Burrell\\ ; Graham Campbell\\ ; Raiyan Chowdhury\\ ; Reinis Cirpons\\ ; Ashley Clayton\\ ; Tom Conti\texttt{\symbol{45}}Leslie\\ ; Joseph Edwards\\ ; Luna Elliott\\ ; Jan Engelhardt\\ ; Isuru Fernando\\ ; Ewan Gilligan\\ ; Gillis Frankie\\ ; Sebastian Gutsche\\ ; Samantha Harper\\ ; Max Horn\\ ; Harry Jack\\ ; Christopher Jefferson\\ ; Malachi Johns\\ ; Olexandr Konovalov\\ ; Hyeokjun Kwon\\ ; Aidan Lau\\ ; Andrea Lee\\ ; Saffron McIver\\ ; Seyyed Ali Mohammadiyeh\\ ; Michael Orlitzky\\ ; Matthew Pancer\\ ; Markus Pfeiffer\\ ; Daniel Pointon\\ ; Pramoth Ragavan\\ ; Lea Racine\\ ; Christopher Russell\\ ; Artur Schaefer\\ ; Isabella Scott\\ ; Kamran Sharma\\ ; Finn Smith\\ ; Ben Spiers\\ ; Maria Tsalakou\\ ; Agastyaa Vishvanath\\ ; Meike Weiss\\ ; Murray Whyte\\ ; Fabian Zickgraf\\ } \end{center}\vfill \mbox{}\\ {\mbox{}\\ \small \noindent \textbf{ Jan De Beule\\ } Email: \href{mailto://jdebeule@cage.ugent.be} {\texttt{jdebeule@cage.ugent.be}}\\ Homepage: \href{https://researchportal.vub.be/en/persons/jan-de-beule} {\texttt{ht Address: \begin{minipage}[t]{8cm}\noindent Vrije Universiteit Brussel, Vakgroep Wiskunde, Pleinlaan 2, B \texttt{\symbol{45}} 1050 Brussels, Belgium\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Julius Jonusas\\ } Email: \href{mailto://j.jonusas@gmail.com} {\texttt{j.jonusas@gmail.com}}\\ Homepage: \href{http://julius.jonusas.work} {\texttt{http://julius.jonusas.work}}} {\mbox{}\\ \small \noindent \textbf{ James Mitchell\\ } Email: \href{mailto://jdm3@st-andrews.ac.uk} {\texttt{jdm3@st\texttt{\symbol{45}} Homepage: \href{https://jdbm.me} {\texttt{https://jdbm.me}}\\ Address: \begin{minipage}[t]{8cm}\noindent Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Wilf A. Wilson\\ } Email: \href{mailto://gap@wilf-wilson.net} {\texttt{gap@wilf\texttt{\symbol{45}}w Homepage: \href{https://wilf.me} {\texttt{https://wilf.me}}}\\ {\mbox{}\\ \small \noindent \textbf{ Michael Young\\ } Email: \href{mailto://mct25@st-andrews.ac.uk} {\texttt{mct25@st\texttt{\symbol{45 Homepage: \href{https://myoung.uk/work/} {\texttt{https://myoung.uk/work/}}\\ Address: \begin{minipage}[t]{8cm}\noindent Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Marina Anagnostopoulou\texttt{\symbol{45}}Merkouri\\ } Email: \href{mailto://mam49@st-andrews.ac.uk} {\texttt{mam49@st\texttt{\symbol{45 Address: \begin{minipage}[t]{8cm}\noindent Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Finn Buck\\ } Email: \href{mailto://finneganlbuck@gmail.com} {\texttt{finneganlbuck@gmail.com}} Address: \begin{minipage}[t]{8cm}\noindent Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Stuart Burrell\\ } Email: \href{mailto://stuartburrell1994@gmail.com} {\texttt{stuartburrell1994@gma Homepage: \href{https://stuartburrell.github.io} {\texttt{https://stuartburrell.gi {\mbox{}\\ \small \noindent \textbf{ Reinis Cirpons\\ } Email: \href{mailto://rc234@st-andrews.ac.uk} {\texttt{rc234@st\texttt{\symbol{45 Address: \begin{minipage}[t]{8cm}\noindent Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Ashley Clayton\\ } Email: \href{mailto://ac323@st-andrews.ac.uk} {\texttt{ac323@st\texttt{\symbol{45 Address: \begin{minipage}[t]{8cm}\noindent Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Tom Conti\texttt{\symbol{45}}Leslie\\ } Email: \href{mailto://tom.contileslie@gmail.com} {\texttt{tom.contileslie@gmail.c Homepage: \href{https://tomcontileslie.com} {\texttt{https://tomcontileslie.com}}} {\mbox{}\\ \small \noindent \textbf{ Joseph Edwards\\ } Email: \href{mailto://jde1@st-andrews.ac.uk} {\texttt{jde1@st\texttt{\symbol{45}} Homepage: \href{https://github.com/Joseph-Edwards} {\texttt{https://github.com/Jos Address: \begin{minipage}[t]{8cm}\noindent Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Luna Elliott\\ } Email: \href{mailto://luna.elliott142857@gmail.com} {\texttt{luna.elliott142857@g Homepage: \href{https://research.manchester.ac.uk/en/persons/luna-elliott} {\textt {\mbox{}\\ \small \noindent \textbf{ Jan Engelhardt\\ } Email: \href{mailto://jengelh@inai.de} {\texttt{jengelh@inai.de}}\\ Homepage: \href{https://inai.de} {\texttt{https://inai.de}}}\\ {\mbox{}\\ \small \noindent \textbf{ Isuru Fernando\\ } Email: \href{mailto://isuruf@gmail.com} {\texttt{isuruf@gmail.com}}}\\ {\mbox{}\\ \small \noindent \textbf{ Ewan Gilligan\\ } Email: \href{mailto://eg207@st-andrews.ac.uk} {\texttt{eg207@st\texttt{\symbol{45 {\mbox{}\\ \small \noindent \textbf{ Gillis Frankie\\ } Email: \href{mailto://fotg1@st-andrews.ac.uk} {\texttt{fotg1@st\texttt{\symbol{45 {\mbox{}\\ \small \noindent \textbf{ Sebastian Gutsche\\ } Email: \href{mailto://gutsche@momo.math.rwth-aachen.de} {\texttt{gutsche@momo.mat {\mbox{}\\ \small \noindent \textbf{ Samantha Harper\\ } Email: \href{mailto://seh25@st-andrews.ac.uk} {\texttt{seh25@st\texttt{\symbol{45 {\mbox{}\\ \small \noindent \textbf{ Max Horn\\ } Email: \href{mailto://mhorn@rptu.de} {\texttt{mhorn@rptu.de}}\\ Homepage: \href{https://www.quendi.de/math} {\texttt{https://www.quendi.de/math}}\ Address: \begin{minipage}[t]{8cm}\noindent Fachbereich Mathematik, RPTU Kaiserslautern\texttt{\symbol{45}}Landau, Gottlieb\texttt{\symbol{45}}Daimler\texttt{\symbol{45}}Stra{\ss}e 48, 67663 Kaiserslautern, Germany\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Harry Jack\\ } Email: \href{mailto://hrj4@st-andrews.ac.uk} {\texttt{hrj4@st\texttt{\symbol{45}} {\mbox{}\\ \small \noindent \textbf{ Christopher Jefferson\\ } Email: \href{mailto://caj21@st-andrews.ac.uk} {\texttt{caj21@st\texttt{\symbol{45 Homepage: \href{https://heather.cafe/} {\texttt{https://heather.cafe/}}\\ Address: \begin{minipage}[t]{8cm}\noindent Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Malachi Johns\\ } Email: \href{mailto://zlj1@st-andrews.ac.uk} {\texttt{zlj1@st\texttt{\symbol{45}} {\mbox{}\\ \small \noindent \textbf{ Olexandr Konovalov\\ } Email: \href{mailto://obk1@st-andrews.ac.uk} {\texttt{obk1@st\texttt{\symbol{45}} Homepage: \href{https://olexandr-konovalov.github.io/} {\texttt{https://olexandr\t Address: \begin{minipage}[t]{8cm}\noindent Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Hyeokjun Kwon\\ } Email: \href{mailto://hk78@st-andrews.ac.uk} {\texttt{hk78@st\texttt{\symbol{45}} {\mbox{}\\ \small \noindent \textbf{ Andrea Lee\\ } Email: \href{mailto://ahwl1@st-andrews.ac.uk} {\texttt{ahwl1@st\texttt{\symbol{45 {\mbox{}\\ \small \noindent \textbf{ Saffron McIver\\ } Email: \href{mailto://sm544@st-andrews.ac.uk} {\texttt{sm544@st\texttt{\symbol{45 {\mbox{}\\ \small \noindent \textbf{ Seyyed Ali Mohammadiyeh\\ } Email: \href{mailto://MaxBaseCode@Gmail.Com} {\texttt{MaxBaseCode@Gmail.Com}}}\\ {\mbox{}\\ \small \noindent \textbf{ Michael Orlitzky\\ } Email: \href{mailto://michael@orlitzky.com} {\texttt{michael@orlitzky.com}}\\ Homepage: \href{https://michael.orlitzky.com/} {\texttt{https://michael.orlitzky.c {\mbox{}\\ \small \noindent \textbf{ Matthew Pancer\\ } Email: \href{mailto://mp322@st-andrews.ac.uk} {\texttt{mp322@st\texttt{\symbol{45 {\mbox{}\\ \small \noindent \textbf{ Markus Pfeiffer\\ } Email: \href{mailto://markus.pfeiffer@morphism.de} {\texttt{markus.pfeiffer@morph Homepage: \href{https://markusp.morphism.de/} {\texttt{https://markusp.morphism.de {\mbox{}\\ \small \noindent \textbf{ Daniel Pointon\\ } Email: \href{mailto://dp211@st-andrews.ac.uk} {\texttt{dp211@st\texttt{\symbol{45 {\mbox{}\\ \small \noindent \textbf{ Pramoth Ragavan\\ } Email: \href{mailto://107881923+pramothragavan@users.noreply.github.com} {\texttt {\mbox{}\\ \small \noindent \textbf{ Lea Racine\\ } Email: \href{mailto://lr217@st-andrews.ac.uk} {\texttt{lr217@st\texttt{\symbol{45 Address: \begin{minipage}[t]{8cm}\noindent Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Artur Schaefer\\ } Email: \href{mailto://as305@st-and.ac.uk} {\texttt{as305@st\texttt{\symbol{45}}an {\mbox{}\\ \small \noindent \textbf{ Isabella Scott\\ } Email: \href{mailto://iscott@uchicago.edu} {\texttt{iscott@uchicago.edu}}}\\ {\mbox{}\\ \small \noindent \textbf{ Kamran Sharma\\ } Email: \href{mailto://kks4@st-andrews.ac.uk} {\texttt{kks4@st\texttt{\symbol{45}} Address: \begin{minipage}[t]{8cm}\noindent Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Finn Smith\\ } Email: \href{mailto://fls3@st-andrews.ac.uk} {\texttt{fls3@st\texttt{\symbol{45}} Address: \begin{minipage}[t]{8cm}\noindent Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Ben Spiers\\ } Email: \href{mailto://bspiers972@outlook.com} {\texttt{bspiers972@outlook.com}}}\ {\mbox{}\\ \small \noindent \textbf{ Maria Tsalakou\\ } Email: \href{mailto://mt200@st-andrews.ac.uk} {\texttt{mt200@st\texttt{\symbol{45 Homepage: \href{https://mariatsalakou.github.io/} {\texttt{https://mariatsalakou.g Address: \begin{minipage}[t]{8cm}\noindent Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Agastyaa Vishvanath\\ } Email: \href{mailto://av215@st-andrews.ac.uk} {\texttt{av215@st\texttt{\symbol{45 {\mbox{}\\ \small \noindent \textbf{ Meike Weiss\\ } Email: \href{mailto://weiss@art.rwth-aachen.de} {\texttt{weiss@art.rwth\texttt{\s Homepage: \href{https://bit.ly/4e6pUeP} {\texttt{https://bit.ly/4e6pUeP}}\\ Address: \begin{minipage}[t]{8cm}\noindent Chair of Algebra and Representation Theory, Pontdriesch 10\texttt{\symbol{45}}16, 52062 Aachen\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Murray Whyte\\ } Email: \href{mailto://mw231@st-andrews.ac.uk} {\texttt{mw231@st\texttt{\symbol{45 Address: \begin{minipage}[t]{8cm}\noindent Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Fabian Zickgraf\\ } Email: \href{mailto://f.zickgraf@dashdos.com} {\texttt{f.zickgraf@dashdos.com}}}\ \end{titlepage} \newpage\setcounter{page}{2} {\small \section*{Abstract} \logpage{[ 0, 0, 1 ]} The \textsf{Digraphs} package is a \textsf{GAP} package containing methods for graphs, digr {\small \section*{Copyright} \logpage{[ 0, 0, 2 ]} Jan De Beule, Julius Jonu{\v s}as, James D. Mitchell, Wilf A. Wilson, Michael Young et al. \textsf{Digraphs} is free software; you can redistribute it and/or modify it under the terms o the \href{ https://www.fsf.org/licenses/gpl.html} {GNU General Public License} as publis or (at your option) any later version. \mbox{}}\\[1cm] {\small \section*{Acknowledgements} \logpage{[ 0, 0, 3 ]} We would like to thank Christopher Jefferson for his help in including \href{http://www.tcs. Soicher, and the inspiration of his \href{https://gap-packages.github.io/grape} {GRAPE} by Max Neunh{\"o}ffer, and independently Artur Sch{\"a}fer. \mbox{}}\\[1cm] \newpage \def\contentsname{Contents\logpage{[ 0, 0, 4 ]}} \tableofcontents \newpage \chapter{\textcolor{Chapter }{ The \textsf{Digraphs} package }}\label{The Digraphs pack \logpage{[ 1, 0, 0 ]} \hyperdef{L}{X7F202ABA780E595D}{} { \index{Digraphs@\textsf{Digraphs} package overview} \section{\textcolor{Chapter }{Introduction}}\logpage{[ 1, 1, 0 ]} \hyperdef{L}{X7DFB63A97E67C0A1}{} { This is the manual for version 1.13.1 of the \textsf{Digraphs} package. This package was devel \begin{itemize} \item Jan De Beule, \item Julius Jonu{\v s}as, \item James D. Mitchell, \item Maria Tsalakou, \item Wilf A. Wilson, and \item Michael C. Young. \end{itemize} Additional contributions were made by many people, for which the authors are very grateful. A full list can be found on the title page. The \textsf{Digraphs} package contai mutable and immutable digraphs and computing information about them. Full explanations of all the functions contained in the package are provided below. If the \href{https://gap-packages.github.io/grape} {GRAPE} package is available, it will the \textsf{Digraphs} package can be converted to \href{https://gap-packages.github.io/ The \href{http://www.tcs.tkk.fi/Software/bliss/} {bliss} tool \cite{JK07} is included i computing automorphism groups and canonical forms of graphs, written by Tommi Junttila and Petteri Kaski. Several of the methods in the \textsf{Digraphs} package rely on \h The \href{https://github.com/graph-algorithms/edge-addition-planarity-suite} {edge planar graph embedding algorithm and related algorithms by John M. Boyer. See Section \ref{Planarity} for more details. From version 1.0.0 of this package, digraphs can be either mutable or immutable. Mutable digraphs can be changed in\texttt{\symbol{45}}place by many of the methods in the package, which avoids unnecessary copying. Immutable digraphs cannot be changed in\texttt{\symbol{45}}place, but their advantage is that the value of an attribute of an immutable digraph is only ever computed once. Mutable digraphs can be converted into immutable digraphs in\texttt{\symbol{45}}place using \texttt{MakeImmutable} (\textbf{Reference: MakeImm that in practice the authors often wanted to create a digraph and immediately modify it (removing certain edges, loops, and so on). Before version 1.0.0, this involved copying the digraph several times, with each copy being discarded almost immediately. From version 1.0.0, this unnecessary copying can be eliminated by first creating a mutable digraph, then changing it in\texttt{\symbol{45}}place, and finally converting the mutable digraph to an immutable one in\texttt{\symbol{45}}place (if desirable). \subsection{\textcolor{Chapter }{Definitions}}\label{Definitions} \logpage{[ 1, 1, 1 ]} \hyperdef{L}{X84541F61810C741D}{} { For the purposes of this package and its documentation, the following definitions apply: A \emph{digraph} $E=(E^0,E^1,r,s)$, also known as a \emph{directed graph}, consists of a set A \emph{directed walk} on a digraph is a sequence of alternating vertices and edges $(v_1, e_1, The \emph{length} of a directed walk $(v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)$ is equal to $n-1$ is the empty directed walk $()$. A \emph{simple circuit} is another name for a non\texttt{\sym non\texttt{\symbol{45}}empty directed cycle. } } } \chapter{\textcolor{Chapter }{Installing \textsf{Digraphs}}}\label{Installing Digra \logpage{[ 2, 0, 0 ]} \hyperdef{L}{X817088B27F90D596}{} { \section{\textcolor{Chapter }{For those in a hurry}}\label{For those in a hurry} \logpage{[ 2, 1, 0 ]} \hyperdef{L}{X7DA3059C79842BF3}{} { In this section we give a brief description of how to start using \textsf{Digraphs}. It is assumed that you have a working copy of \textsf{GAP} with version number 4.10.0 or higher. up\texttt{\symbol{45}}to\texttt{\symbol{45}}date version of \textsf{GAP} and instruct The following is a summary of the steps that should lead to a successful installation of \textsf{Digraphs}: \begin{itemize} \item ensure that the \href{https://gap-packages.github.io/io} {IO} package version 4.5. \item ensure that the \textsf{orb} package version 4.8.2 or higher is available. \textsf{orb it is not required to be compiled for \textsf{Digraphs} to be loaded. \item ensure that the \textsf{datastructures} package version 0.2.5 or higher is available. \item \textsc{This step is optional:} certain functions in \textsf{Digraphs} require the \h Subsection \ref{The Grape package} will not work. \item \textsc{This step is optional:} certain functions in \textsf{Digraphs} require the \h \item download the package archive \texttt{digraphs\texttt{\symbol{45}}1.13.1.tar.gz. \item unzip and untar the file, this should create a directory called \texttt{digraphs\textt \item locate the \texttt{pkg} directory of your \textsf{GAP} directory, which contains the d \item it is necessary to compile the \textsf{Digraphs} package. Inside the \texttt{pkg/digr \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=] ./configure make \end{Verbatim} Further information about this step can be found in Section \ref{Compiling the kernel module} \item start \textsf{GAP} in the usual way (i.e. type \texttt{gap} at the command line). \item type \texttt{LoadPackage("digraphs");} \end{itemize} If you want to check that the package is working correctly, you should run some of the tests described in Section \ref{Testing your installation}. \subsection{\textcolor{Chapter }{Configuration options}}\label{Configuration option \logpage{[ 2, 1, 1 ]} \hyperdef{L}{X7C1F2E6D860DBDEC}{} { In addition to the usual autoconf generated configuration flags, the following flags are provided. \begin{center} \begin{tabular}{|l|l|}\hline Option& Meaning\\ \hline \hline \texttt{\texttt{\symbol{45}}\texttt{\symbol{45}}enable\texttt{\symbol{45}}code\t enable code coverage support\\ \texttt{\texttt{\symbol{45}}\texttt{\symbol{45}}enable\texttt{\symbol{45}}compil enable compiler warnings\\ \texttt{\texttt{\symbol{45}}\texttt{\symbol{45}}enable\texttt{\symbol{45}}debug}& enable debug mode\\ \texttt{\texttt{\symbol{45}}\texttt{\symbol{45}}with\texttt{\symbol{45}}external use external \href{http://www.tcs.tkk.fi/Software/bliss/} {bliss} \\ \texttt{\texttt{\symbol{45}}\texttt{\symbol{45}}with\texttt{\symbol{45}}external use external \href{https://github.com/graph-algorithms/edge-addition-planarity-sui \texttt{\texttt{\symbol{45}}\texttt{\symbol{45}}with\texttt{\symbol{45}}gaproot}& specify root of GAP installation\\ \texttt{\texttt{\symbol{45}}\texttt{\symbol{45}}without\texttt{\symbol{45}}intri do not use compiler intrinsics even if available\\ \hline \end{tabular}\\[2mm] \textbf{Table: }Configuration flags\end{center} } } \section{\textcolor{Chapter }{Optional package dependencies}}\label{Optional package \logpage{[ 2, 2, 0 ]} \hyperdef{L}{X780AA9D97EBCA95D}{} { The \textsf{Digraphs} package is written in \textsf{GAP} and C code and requires the \href{ht bipartitions to a file. \subsection{\textcolor{Chapter }{The Grape package}}\label{The Grape package} \logpage{[ 2, 2, 1 ]} \hyperdef{L}{X8493C7587FCF6D8B}{} { The \href{https://gap-packages.github.io/grape} {GRAPE} package must be available for th \begin{itemize} \item \texttt{Graph} (\ref{Graph}) with a digraph argument \item \texttt{AsGraph} (\ref{AsGraph}) with a digraph argument \item \texttt{Digraph} (\ref{Digraph}) with a \href{https://gap-packages.github.io/gr \end{itemize} If \href{https://gap-packages.github.io/grape} {GRAPE} is not available, then \textsf{D work. } } \section{\textcolor{Chapter }{Compiling the kernel module}}\label{Compiling the kernel \logpage{[ 2, 3, 0 ]} \hyperdef{L}{X849F6196875A6DF5}{} { The \textsf{Digraphs} package has a \textsf{GAP} kernel component in C which should be compil certain low\texttt{\symbol{45}}level functions required by \textsf{Digraphs}. It is not possible to use the \textsf{Digraphs} package without compiling it. To compile the kernel component inside the \texttt{pkg/digraphs\texttt{\symbol{45}}1.13 \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=] ./configure make \end{Verbatim} If you installed the package in another 'pkg' directory than the standard 'pkg' directory in your \textsf{GAP} installation, then you have to do two things. Firstly dur have to use the option '\texttt{\symbol{45}}\texttt{\symbol{45}}with\texttt{\symbol{45}}gaproot=PATH' of the 'configure' script where 'PATH' is a path to the main GAP root directory (if not given the default '../..' is assumed). If you installed \textsf{GAP} on several architectures, you must execute the configure/make the architectures. You can either do this immediately after configuring and compiling GAP itself on this architecture, or alternatively set the environment variable 'CONFIGNAME' to the name of the configuration you used when compiling GAP before running './configure'. Note however that your compiler choice and flags (environment variables 'CC' and 'CFLAGS') need to be chosen to match the setup of the original GAP compilation. For example you have to specify 32\texttt{\symbol{45}}bit or 64\texttt{\symbol{45}}bit mode correctly! } \section{\textcolor{Chapter }{Rebuilding the documentation}}\label{Rebuilding the doc \logpage{[ 2, 4, 0 ]} \hyperdef{L}{X857CBE5484CF703A}{} { The \textsf{Digraphs} package comes complete with pdf, html, and text versions of the documen However, you might find it necessary, at some point, to rebuild the documentation. To rebuild the documentation, please use the function \texttt{DigraphsMake \subsection{\textcolor{Chapter }{DigraphsMakeDoc}} \logpage{[ 2, 4, 1 ]}\nobreak \hyperdef{L}{X870631C38610AC25}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DigraphsMakeDoc( \label{DigraphsMakeDoc} }\hfill{\scriptsize (function)}}\\ \textbf{\indent Returns:\ } Nothing This function should be called with no argument to compile the \textsf{Digraphs} documentati } \section{\textcolor{Chapter }{Testing your installation}}\label{Testing your installa \logpage{[ 2, 5, 0 ]} \hyperdef{L}{X7862D3F37C5BBDEF}{} { In this section we describe how to test that \textsf{Digraphs} is working as intended. To test t If something goes wrong, then please review the instructions in Section \ref{For those in a hur \subsection{\textcolor{Chapter }{DigraphsTestInstall}} \logpage{[ 2, 5, 1 ]}\nobreak \hyperdef{L}{X86AF4DAE80B978DA}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DigraphsTestInst \label{DigraphsTestInstall} }\hfill{\scriptsize (function)}}\\ \textbf{\indent Returns:\ } \texttt{true} or \texttt{false}. This function can be called without arguments to test your installation of \textsf{Digraphs} complete. To test more comprehensively that \textsf{Digraphs} is working correctly, use \te \subsection{\textcolor{Chapter }{DigraphsTestStandard}} \logpage{[ 2, 5, 2 ]}\nobreak \hyperdef{L}{X7A20088B7C406C4A}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DigraphsTestStan \label{DigraphsTestStandard} }\hfill{\scriptsize (function)}}\\ \textbf{\indent Returns:\ } \texttt{true} or \texttt{false}. This function can be called without arguments to test all of the methods included in \textsf{Digraphs}. These tests should take less than a minute to complete. To quickly test that \textsf{Digraphs} is installed correctly use \texttt{DigraphsTestIns \subsection{\textcolor{Chapter }{DigraphsTestExtreme}} \logpage{[ 2, 5, 3 ]}\nobreak \hyperdef{L}{X7885EF3E785D4298}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DigraphsTestExtr \label{DigraphsTestExtreme} }\hfill{\scriptsize (function)}}\\ \textbf{\indent Returns:\ } \texttt{true} or \texttt{false}. This function should be called with no argument. It executes a series of very demanding tests, which measure the performance of a variety of functions on large examples. These tests take a long time to complete, at least several minutes. For these tests to complete, the digraphs library \texttt{digraphs\texttt{\symbol{45}}li } } \chapter{\textcolor{Chapter }{Creating digraphs}}\label{Creating digraphs} \logpage{[ 3, 0, 0 ]} \hyperdef{L}{X7D34861E863A5D93}{} { In this chapter we describe how to create digraphs. \section{\textcolor{Chapter }{Creating digraphs}}\logpage{[ 3, 1, 0 ]} \hyperdef{L}{X7D34861E863A5D93}{} { \subsection{\textcolor{Chapter }{IsDigraph}} \logpage{[ 3, 1, 1 ]}\nobreak \hyperdef{L}{X7877ADC77F85E630}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsDigraph\index{ \label{IsDigraph} }\hfill{\scriptsize (Category)}}\\ Every digraph in \textsf{Digraphs} belongs to the category \texttt{IsDigraph}. Some basic a \subsection{\textcolor{Chapter }{IsMutableDigraph}} \logpage{[ 3, 1, 2 ]}\nobreak \hyperdef{L}{X7D7EDF83820ED6F5}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsMutableDigraph \label{IsMutableDigraph} }\hfill{\scriptsize (Category)}}\\ \texttt{IsMutableDigraph} is a synonym for \texttt{IsDigraph} (\ref{IsDigraph}) and \tex the \textsf{Digraphs} package, and is not attribute\texttt{\symbol{45}}storing {\texten A mutable digraph may be converted into an immutable attribute\texttt{\symbol{45}}storing digraph by calling \texttt{MakeImmutable} (\text \subsection{\textcolor{Chapter }{IsImmutableDigraph}} \logpage{[ 3, 1, 3 ]}\nobreak \hyperdef{L}{X7CAFAA89804F80BD}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsImmutableDigra \label{IsImmutableDigraph} }\hfill{\scriptsize (Category)}}\\ \texttt{IsImmutableDigraph} is a subcategory of \texttt{IsDigraph} (\ref{IsDigraph}). D lie in \texttt{IsAttributeStoringRep} (\textbf{Reference: IsAttributeStoringRep}). A mutable digraph may be converted to an immutable digraph that lies in the category \texttt{IsImmutableDigraph} by calling \texttt{MakeImmutable} (\textbf{Refer The operation \texttt{DigraphMutableCopy} (\ref{DigraphMutableCopy}) can be used to cons not possible to convert an immutable digraph into a mutable digraph in\texttt{\symbol{45}}place. } \subsection{\textcolor{Chapter }{IsCayleyDigraph}} \logpage{[ 3, 1, 4 ]}\nobreak \hyperdef{L}{X7E749324800B38A5}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsCayleyDigraph\ \label{IsCayleyDigraph} }\hfill{\scriptsize (Category)}}\\ \texttt{IsCayleyDigraph} is a subcategory of \texttt{IsDigraph}. Digraphs that are Cayley operation \texttt{CayleyDigraph} (\ref{CayleyDigraph}) are constructed in this category \subsection{\textcolor{Chapter }{IsDigraphWithAdjacencyFunction}} \logpage{[ 3, 1, 5 ]}\nobreak \hyperdef{L}{X80F1B6D28478D8B9}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsDigraphWithAdj \label{IsDigraphWithAdjacencyFunction} }\hfill{\scriptsize (Category)}}\\ \texttt{IsDigraphWithAdjacencyFunction} is a subcategory of \texttt{IsDigraph}. Digrap \subsection{\textcolor{Chapter }{DigraphByOutNeighboursType}} \logpage{[ 3, 1, 6 ]}\nobreak \hyperdef{L}{X86E798B779515678}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DigraphByOutNeig \label{DigraphByOutNeighboursType} }\hfill{\scriptsize (global variable)}}\\ \noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DigraphFamily\ind \label{DigraphFamily} }\hfill{\scriptsize (family)}}\\ The type of all digraphs is \texttt{DigraphByOutNeighboursType}. The family of all digraphs \subsection{\textcolor{Chapter }{Digraph}} \logpage{[ 3, 1, 7 ]}\nobreak \hyperdef{L}{X834843057CE86655}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{Digraph({\mdseri \label{Digraph} }\hfill{\scriptsize (operation)}}\\ \noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{Digraph({\mdserie \label{Digraph:for a list and function} }\hfill{\scriptsize (operation)}}\\ \noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{Digraph({\mdserie \label{Digraph:for a group, list, function, and function} }\hfill{\scriptsize (operation)}}\\ \textbf{\indent Returns:\ } A digraph. If the optional first argument \mbox{\texttt{\mdseries\slshape filt}} is present, then thi digraph being created will belong to. For example, if \mbox{\texttt{\mdseries\slshape filt \begin{description} \item[{for a list (i.e. an adjacency list)}] if \mbox{\texttt{\mdseries\slshape obj}} is a l More precisely, there is an edge from vertex \texttt{i} to \texttt{j} if and only if \texttt{j} \item[{for three lists}] if \mbox{\texttt{\mdseries\slshape obj}} is a duplicate\texttt{ The vertices of the digraph will be labelled by the elements of \mbox{\texttt{\mdseries\slsh \item[{for an integer, and two lists}] if \mbox{\texttt{\mdseries\slshape obj}} is an integ \item[{for a list and a function}] if \mbox{\texttt{\mdseries\slshape list}} is a list and \m \item[{for a group, a list, and two functions}] The arguments will be \mbox{\texttt{\mdserie Let \mbox{\texttt{\mdseries\slshape G}} be a group acting on the objects in \mbox{\texttt{\ of \mbox{\texttt{\mdseries\slshape list}} and directed edges \texttt{[x, y]} when \texttt The action of the group \mbox{\texttt{\mdseries\slshape G}} on the objects in \mbox{\texttt \item[{for a Grape package graph}] if \mbox{\texttt{\mdseries\slshape obj}} is a \href{htt \item[{for a binary relation}] if \mbox{\texttt{\mdseries\slshape obj}} is a binary relati \item[{for a string naming a digraph}] if \mbox{\texttt{\mdseries\slshape obj}} is a non\te digraph that has name \mbox{\texttt{\mdseries\slshape obj}}. \textsf{Digraphs} comes wit be loaded in this way. Valid names include \texttt{"folkman"}, \texttt{"diamond"} and \text Note that any undirected graphs in the database are stored as symmetric digraphs, so the resulting digraph will have twice as many edges as its undirected counterpart. \end{description} \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@gr := Digraph([| !gapprompt@>| !gapinput@[2, 5, 8, 10], [2, 3, 4, 2, 5, 6, 8, 9, 10], [1],| !gapprompt@>| !gapinput@[3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],| !gapprompt@>| !gapinput@[1, 5, 9], [1, 2, 7, 8], [3, 5]]);| <immutable multidigraph with 10 vertices, 38 edges> !gapprompt@gap>| !gapinput@gr := Digraph(["a", "b", "c"], ["a"], ["b"]);| <immutable digraph with 3 vertices, 1 edge> !gapprompt@gap>| !gapinput@gr := Digraph(5, [1, 2, 2, 4, 1, 1], [2, 3, 5, 5, 1, 1]);| <immutable multidigraph with 5 vertices, 6 edges> !gapprompt@gap>| !gapinput@Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,| !gapprompt@>| !gapinput@function(x, y) return Intersection(x, y) = []; end);;| !gapprompt@gap>| !gapinput@Digraph(Petersen);| <immutable digraph with 10 vertices, 30 edges> !gapprompt@gap>| !gapinput@gr := Digraph([1 .. 10], ReturnTrue);| <immutable digraph with 10 vertices, 100 edges> !gapprompt@gap>| !gapinput@Digraph("Diamond");| <immutable digraph with 4 vertices, 10 edges> \end{Verbatim} The next example illustrates the uses of the fourth and fifth variants of this constructor. The resulting digraph is a strongly regular graph, and it is actually the point graph of the van Lint\texttt{\symbol{45}}Schrijver partial geometry, \cite{vLS81}. The algebraic description is taken from the seminal paper of Calderb Kantor \cite{CK86}. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@f := GF(3 ^ 4);| GF(3^4) !gapprompt@gap>| !gapinput@gamma := First(f, x -> Order(x) = 5);| Z(3^4)^64 !gapprompt@gap>| !gapinput@L := Union([Zero(f)], List(Group(gamma)));| [ 0*Z(3), Z(3)^0, Z(3^4)^16, Z(3^4)^32, Z(3^4)^48, Z(3^4)^64 ] !gapprompt@gap>| !gapinput@omega := Union(List(L, x -> List(Difference(L, [x]), y -> x - y)));| [ Z(3)^0, Z(3), Z(3^4)^5, Z(3^4)^7, Z(3^4)^8, Z(3^4)^13, Z(3^4)^15, Z(3^4)^16, Z(3^4)^21, Z(3^4)^23, Z(3^4)^24, Z(3^4)^29, Z(3^4)^31, Z(3^4)^32, Z(3^4)^37, Z(3^4)^39, Z(3^4)^45, Z(3^4)^47, Z(3^4)^48, Z(3^4)^53, Z(3^4)^55, Z(3^4)^56, Z(3^4)^61, Z(3^4)^63, Z(3^4)^64, Z(3^4)^69, Z(3^4)^71, Z(3^4)^72, Z(3^4)^77, Z(3^4)^79 ] !gapprompt@gap>| !gapinput@adj := function(x, y)| !gapprompt@>| !gapinput@ return x - y in omega;| !gapprompt@>| !gapinput@end;| function( x, y ) ... end !gapprompt@gap>| !gapinput@digraph := Digraph(AsList(f), adj);| <immutable digraph with 81 vertices, 2430 edges> !gapprompt@gap>| !gapinput@group := Group(Z(3));;| !gapprompt@gap>| !gapinput@act := \*;| <Operation "*"> !gapprompt@gap>| !gapinput@digraph := Digraph(group, List(f), act, adj);| <immutable digraph with 81 vertices, 2430 edges> \end{Verbatim} } \subsection{\textcolor{Chapter }{DigraphByAdjacencyMatrix}} \logpage{[ 3, 1, 8 ]}\nobreak \hyperdef{L}{X8023FE387A3AB609}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DigraphByAdjacen \label{DigraphByAdjacencyMatrix} }\hfill{\scriptsize (operation)}}\\ \textbf{\indent Returns:\ } A digraph. If the optional first argument \mbox{\texttt{\mdseries\slshape filt}} is present, then thi digraph being created will belong to. For example, if \mbox{\texttt{\mdseries\slshape filt If \mbox{\texttt{\mdseries\slshape list}} is the adjacency matrix of a digraph in the sense o Alternatively, if \mbox{\texttt{\mdseries\slshape list}} is a square boolean matrix, then \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@DigraphByAdjacencyMatrix([| !gapprompt@>| !gapinput@[0, 1, 0, 2, 0],| !gapprompt@>| !gapinput@[1, 1, 1, 0, 1],| !gapprompt@>| !gapinput@[0, 3, 2, 1, 1],| !gapprompt@>| !gapinput@[0, 0, 1, 0, 1],| !gapprompt@>| !gapinput@[2, 0, 0, 0, 0]]);| <immutable multidigraph with 5 vertices, 18 edges> !gapprompt@gap>| !gapinput@D := DigraphByAdjacencyMatrix([| !gapprompt@>| !gapinput@[true, false, true],| !gapprompt@>| !gapinput@[false, false, true],| !gapprompt@>| !gapinput@[false, true, false]]);| <immutable digraph with 3 vertices, 4 edges> !gapprompt@gap>| !gapinput@OutNeighbours(D);| [ [ 1, 3 ], [ 3 ], [ 2 ] ] !gapprompt@gap>| !gapinput@D := DigraphByAdjacencyMatrix(IsMutableDigraph, | !gapprompt@>| !gapinput@[[true, false, true],| !gapprompt@>| !gapinput@ [false, false, true],| !gapprompt@>| !gapinput@ [false, true, false]]);| <mutable digraph with 3 vertices, 4 edges> \end{Verbatim} } \subsection{\textcolor{Chapter }{DigraphByEdges}} \logpage{[ 3, 1, 9 ]}\nobreak \hyperdef{L}{X7F37B6768349E269}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DigraphByEdges({ \label{DigraphByEdges} }\hfill{\scriptsize (operation)}}\\ \textbf{\indent Returns:\ } A digraph. If the optional first argument \mbox{\texttt{\mdseries\slshape filt}} is present, then thi digraph being created will belong to. For example, if \mbox{\texttt{\mdseries\slshape filt If \mbox{\texttt{\mdseries\slshape list}} is list of pairs of positive integers, then this f with the minimum number of vertices \texttt{m} such that its list equal \mbox{\texttt{\mdser If the optional second argument \mbox{\texttt{\mdseries\slshape n}} is a positive integer w See \texttt{DigraphEdges} (\ref{DigraphEdges}). \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@DigraphByEdges(| !gapprompt@>| !gapinput@[[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],| !gapprompt@>| !gapinput@ [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]]);| <immutable digraph with 6 vertices, 10 edges> !gapprompt@gap>| !gapinput@DigraphByEdges(| !gapprompt@>| !gapinput@[[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],| !gapprompt@>| !gapinput@ [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]], 12);| <immutable digraph with 12 vertices, 10 edges> !gapprompt@gap>| !gapinput@DigraphByEdges(IsMutableDigraph, | !gapprompt@>| !gapinput@[[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],| !gapprompt@>| !gapinput@ [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]], 12);| <mutable digraph with 12 vertices, 10 edges> \end{Verbatim} } \subsection{\textcolor{Chapter }{EdgeOrbitsDigraph}} \logpage{[ 3, 1, 10 ]}\nobreak \hyperdef{L}{X7B75C1D680757D6F}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{EdgeOrbitsDigrap \label{EdgeOrbitsDigraph} }\hfill{\scriptsize (operation)}}\\ \textbf{\indent Returns:\ } An immutable digraph. If \mbox{\texttt{\mdseries\slshape G}} is a permutation group, \mbox{\texttt{\mdseries\ If the optional third argument \mbox{\texttt{\mdseries\slshape n}} is not present, then the \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@digraph := EdgeOrbitsDigraph(Group((1, 3), (1, 2)(3, 4)),| !gapprompt@>| !gapinput@ [[1, 2], [4, 5]], 5);| <immutable digraph with 5 vertices, 12 edges> !gapprompt@gap>| !gapinput@OutNeighbours(digraph);| [ [ 2, 4, 5 ], [ 1, 3, 5 ], [ 2, 4, 5 ], [ 1, 3, 5 ], [ ] ] !gapprompt@gap>| !gapinput@RepresentativeOutNeighbours(digraph);| [ [ 2, 4, 5 ], [ ] ] \end{Verbatim} } \subsection{\textcolor{Chapter }{DigraphByInNeighbours}} \logpage{[ 3, 1, 11 ]}\nobreak \hyperdef{L}{X81BC49B57EAADEFB}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DigraphByInNeigh \label{DigraphByInNeighbours} }\hfill{\scriptsize (operation)}}\\ \noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DigraphByInNeighb \label{DigraphByInNeighbors} }\hfill{\scriptsize (operation)}}\\ \textbf{\indent Returns:\ } A digraph. If the optional first argument \mbox{\texttt{\mdseries\slshape filt}} is present, then thi digraph being created will belong to. For example, if \mbox{\texttt{\mdseries\slshape filt If \mbox{\texttt{\mdseries\slshape list}} is a list of lists of positive integers list the ra If \texttt{i} occurs in the list \texttt{\mbox{\texttt{\mdseries\slshape list}}[j]} with See \texttt{InNeighbours} (\ref{InNeighbours}). \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@D := DigraphByInNeighbours([| !gapprompt@>| !gapinput@[2, 5, 8, 10], [2, 3, 4, 5, 6, 8, 9, 10],| !gapprompt@>| !gapinput@[1], [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],| !gapprompt@>| !gapinput@[1, 5, 9], [1, 2, 7, 8], [3, 5]]);| <immutable digraph with 10 vertices, 37 edges> !gapprompt@gap>| !gapinput@D := DigraphByInNeighbours([[2, 3, 2], [1], [1, 2, 3]]);| <immutable multidigraph with 3 vertices, 7 edges> !gapprompt@gap>| !gapinput@D := DigraphByInNeighbours(IsMutableDigraph, | !gapprompt@>| !gapinput@ [[2, 3, 2], [1], [1, 2, 3]]);| <mutable multidigraph with 3 vertices, 7 edges> \end{Verbatim} } \subsection{\textcolor{Chapter }{CayleyDigraph}} \logpage{[ 3, 1, 12 ]}\nobreak \hyperdef{L}{X7FCADADC7EC28478}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{CayleyDigraph({\ \label{CayleyDigraph} }\hfill{\scriptsize (operation)}}\\ \textbf{\indent Returns:\ } A digraph. Let \mbox{\texttt{\mdseries\slshape G}} be any group and let \mbox{\texttt{\mdseries\sls The vertices of the digraph correspond to the elements of \mbox{\texttt{\mdseries\slshape G The labels of the vertices \texttt{u}, \texttt{v}, and the edge \texttt{[u, v]} are the corres If the optional first argument \mbox{\texttt{\mdseries\slshape filt}} is present, then thi digraph being created will belong to. For example, if \mbox{\texttt{\mdseries\slshape filt If the optional third argument \mbox{\texttt{\mdseries\slshape gens}} is not present, then The digraph created by this operation belongs to the category \texttt{IsCayleyDigraph} (\re \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@G := DihedralGroup(8);| <pc group of size 8 with 3 generators> !gapprompt@gap>| !gapinput@CayleyDigraph(G);| <immutable digraph with 8 vertices, 24 edges> !gapprompt@gap>| !gapinput@G := DihedralGroup(IsPermGroup, 8);| Group([ (1,2,3,4), (2,4) ]) !gapprompt@gap>| !gapinput@CayleyDigraph(G);| <immutable digraph with 8 vertices, 16 edges> !gapprompt@gap>| !gapinput@digraph := CayleyDigraph(G, [()]);| <immutable digraph with 8 vertices, 8 edges> !gapprompt@gap>| !gapinput@GroupOfCayleyDigraph(digraph) = G;| true !gapprompt@gap>| !gapinput@GeneratorsOfCayleyDigraph(digraph);| [ () ] --> -------------------- --> maximum size reached --> -------------------- ¤ Dauer der Verarbeitung: 0.36 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28