House of Graphs: Introduction what are interesting graphs? - PowerPoint PPT Presentation

House of Graphs: what are interesting graphs? CSD5 House of Graphs: Introduction what are interesting graphs? GraPHedron First Definition of interesting graphs Computational G. Brinkmann 1 , K. Coolsaet 1 , results J. Goedgebeur 1 and H. Mélot 1 , 2 The House of Graphs Perspectives 1 Vakgroep Toegpaste Wiskunde en Informatica, UGent, Belgium 2 Service d’Informatique Théorique, UMons, Belgium 21 July 2010 Computers in Scientific Discovery 5, Sheffield, UK

House of Graphs: Objectives what are interesting graphs? CSD5 Introduction GraPHedron First Definition of interesting graphs Main objectives of the House of Graphs project: Computational results ◮ What make a graph relevant or interesting? The House of ◮ Amongst the large number of non isomorphic graphs, is Graphs Perspectives there a few that can be considered as interesting? ◮ How to share the answers of the two previous questions with researchers?

House of Graphs: Notations what are interesting graphs? CSD5 Introduction GraPHedron First Definition of Definition interesting graphs A graph G = (V,E) : Computational results ◮ set V of nodes ; The House of Graphs ◮ set E of edges . Perspectives Remark Graphs considered : simples and undirected

House of Graphs: Notations what are interesting graphs? CSD5 Introduction Definition GraPHedron A graph invariant is a numerical value, preserved by isomorphism. First Definition of interesting graphs Computational Example results Numbers n of nodes and m of edges. The House of Graphs Perspectives Example n = 4 et m = 5.

House of Graphs: What make a graph relevant of interesting? what are interesting graphs? CSD5 Introduction GraPHedron First Definition of interesting graphs We propose two answers: Computational results ◮ appears useful in the literature or in (static) websites; The House of ◮ is pointed out by a conjecture-making system. Graphs Perspectives Examples: complete graphs, cycles, paths, Petersen graph, Heawood graph (cf. Pisanski’s talk), etc.

House of Graphs: Interesting graphs in the literature or on the web what are interesting graphs? CSD5 Interesting graphs in the literature and on the web: counterexamples; tight graphs; classes of graphs, lists of Introduction graphs, etc. GraPHedron First Definition of interesting graphs Computational results The House of Graphs Perspectives

House of Graphs: Interesting graphs in the literature or on the web what are interesting graphs? CSD5 Interesting graphs in the literature and on the web: counterexamples; tight graphs; classes of graphs, lists of Introduction graphs, etc. GraPHedron First Definition of interesting graphs Examples of books: Computational results ◮ Brandstädt, Le and Spinrad, Graph The House of Graphs classes: a survey (1999) Perspectives ◮ Capobianco, Molluzzo, Examples and Counterexamples in Graph Theory (1978)

House of Graphs: Interesting graphs in the literature or on the web what are interesting graphs? CSD5 Interesting graphs in the literature and on the web: counterexamples; tight graphs; classes of graphs, lists of Introduction graphs, etc. GraPHedron First Definition of interesting graphs Examples of books: Computational results ◮ Brandstädt, Le and Spinrad, Graph The House of Graphs classes: a survey (1999) Perspectives ◮ Capobianco, Molluzzo, Examples and Counterexamples in Graph Theory (1978)

House of Graphs: Interesting graphs in the literature or on the web what are interesting graphs? CSD5 Interesting graphs in the literature and on the web: counterexamples; tight graphs; classes of graphs, lists of Introduction graphs, etc. GraPHedron First Definition of interesting graphs Examples of books: Computational results ◮ Brandstädt, Le and Spinrad, Graph The House of Graphs classes: a survey (1999) Perspectives ◮ Capobianco, Molluzzo, Examples and Counterexamples in Graph Theory (1978) Examples of websites (static lists of graphs): ◮ Brendan McKay ◮ Markus Meringer ◮ Gordon Royle

House of Graphs: Interesting graphs in the literature or on the web what are interesting graphs? CSD5 Interesting graphs in the literature and on the web: counterexamples; tight graphs; classes of graphs, lists of Introduction graphs, etc. GraPHedron First Definition of interesting graphs Examples of books: Computational results ◮ Brandstädt, Le and Spinrad, Graph The House of Graphs classes: a survey (1999) Perspectives ◮ Capobianco, Molluzzo, Examples and Counterexamples in Graph Theory (1978) Examples of websites (static lists of graphs): ◮ Brendan McKay ◮ Markus Meringer ◮ Gordon Royle

House of Graphs: Conjecture-making systems what are interesting graphs? CSD5 Introduction For particular problems (conjectures, set of invariants, inequality GraPHedron of invariants, etc.), graphs are pointed out by conjecture-making First Definition of interesting graphs systems. Computational results Examples: The House of Graphs ◮ AutoGraphiX: extremal graphs; Perspectives ◮ GrInvIn: counterexamples; ◮ Graffiti: counterexamples; ◮ GraPHedron: vertex-graphs (= “conglomerates”, see later); ◮ new version of newGRAPH? see Friday. . . ◮ etc.

House of Graphs: Is there a few graphs that are interesting? what are interesting graphs? CSD5 Introduction GraPHedron First Definition of interesting graphs Computational Amongst the large number of non isomorphic graphs, is there a results few that can be considered as interesting? The House of Graphs Perspectives Our hypothesis: very few graphs can be considered as interesting.

House of Graphs: A first definition of interesting graphs what are interesting graphs? CSD5 Introduction GraPHedron Starting point to obtain (automatically) a first set of interesting First Definition of interesting graphs graphs: use of GraPHedron. Computational results GraPHedron 1 The House of Graphs ◮ Computer assisted and automated conjectures Perspectives ◮ Use a polyhedral approach ◮ Conjectures (inequalities among graph’s invariants) : best possible under some conditions 1 HM, Disc. Appl. Math. 156 (2008), 1875-1891

House of Graphs: GraPHedron’s type of problems what are interesting graphs? CSD5 Introduction GraPHedron Definition First Definition of A problem is defined by I , C and n , where interesting graphs Computational ◮ I = ( f , g ) is a pair of graph’s invariants f and g (excluding results the number of nodes n ); The House of Graphs ◮ C is a particular class of graphs; Perspectives ◮ n is a fixed number of nodes. Problem What are all the best linear inequalities among f and g , valid for all graphs of order n in C ?

House of Graphs: GraPHedron’s type of problems what are interesting graphs? CSD5 Introduction Input: a problem defined by I = ( f , g ) , C and n . GraPHedron First Definition of interesting graphs Output: a polyhedral description (polytope P ) of the problem Computational results The House of P = conv { ( x , y ) | ∃ G = ( V , E ) ∈ C , | V | = n , f ( G ) = x , g ( G ) = y } . Graphs Perspectives Remarks ◮ In this framework, we limit I to 2 invariants (not the case in GraPHedron); ◮ In this talk: C will be either general or connected graphs.

House of Graphs: Polyhedral approach what are interesting graphs? Example: diameter D and number of edges m of connected graphs CSD5 Introduction GraPHedron Example ( n = 4) First Definition of interesting graphs I = ( D , m ) and C = connected Computational results The House of Graphs Perspectives

House of Graphs: Polyhedral approach what are interesting graphs? Example: diameter D and number of edges m of connected graphs CSD5 Introduction GraPHedron 1. generate graphs ∈ C n Example ( n = 4) First Definition of interesting graphs I = ( D , m ) and C = connected Computational results Graphs of C 4 : The House of Graphs Perspectives

House of Graphs: Polyhedral approach what are interesting graphs? Example: diameter D and number of edges m of connected graphs CSD5 Introduction GraPHedron 1. generate graphs ∈ C n Example ( n = 4) First Definition of interesting graphs 2. compute invariants of I I = ( D , m ) and C = connected Computational results Graphs of C 4 : The House of Graphs (2,5) (2,3) (2,4) (1,6) Perspectives (3,3) (2,4) Coordinates (D,m)

House of Graphs: Polyhedral approach what are interesting graphs? Example: diameter D and number of edges m of connected graphs CSD5 Introduction GraPHedron 1. generate graphs ∈ C n Example ( n = 4) First Definition of interesting graphs 2. compute invariants of I I = ( D , m ) and C = connected Computational 3. consider graphs as points results The House of in the space Graphs m Perspectives 6 5 4 3 2 1 1 2 3 D

House of Graphs: Polyhedral approach what are interesting graphs? Example: diameter D and number of edges m of connected graphs CSD5 Introduction GraPHedron 1. generate graphs ∈ C n Example ( n = 4) First Definition of interesting graphs 2. compute invariants of I I = ( D , m ) and C = connected Computational 3. consider graphs as points results The House of in the space Graphs m 4. compute the polytope P 6 Perspectives (convex hull) 5 4 3 2 1 1 2 3 D