Linear inequalities among graph invariants: Using GraPHedron to uncover optimal relationships

Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph vertices, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tuples. By definition, the optimal linear inequalities correspond to the facets of this polytope. They are finite in number, are logically independent, and generate precisely all the linear inequalities valid on the class of graphs. The computer system GraPHedron, developed by some of the authors, is able to produce experimental data about such inequalities for a “small” number of vertices. It greatly helps in conjecturing optimal linear inequalities, which are then hopefully proved for any number of vertices. Two examples are investigated here for the class of connected graphs. First, all the optimal linear inequalities for the stability number and the number of edges are obtained. To this aim, a problem of Ore (1962) related to the Turan Theorem (1941) is solved. Second, several optimal inequalities are established for three invariants: the maximum degree, the irregularity, and the diameter. © 2008 Wiley Periodicals, Inc. NETWORKS, 2008

[1]  Pierre Hansen,et al.  Variable Neighborhood Search for Extremal Graphs. 2. Finding Graphs with Extremal Energy , 1998, J. Chem. Inf. Comput. Sci..

[2]  Pierre Hansen,et al.  Variable neighborhood search for extremal graphs: 1 The AutoGraphiX system , 1997, Discret. Math..

[3]  Siemion Fajtlowicz,et al.  On conjectures of Graffiti , 1988, Discret. Math..

[4]  O. Ore.  Theory of Graphs , 1962 .

[5]  M. Newman.  A result on integral symmetric matrices , 1974 .

[6]  L. Collatz,et al.  Spektren endlicher grafen , 1957 .

[7]  Bela Bollobas,et al.  Graph theory , 1979 .

[8]  Hadrien Mélot,et al.  Facet defining inequalities among graph invariants: The system GraPHedron , 2008, Discret. Appl. Math..

[9]  Ronald D. Dutton,et al.  A compilation of relations between graph invariants , 1985, Networks.

[10]  Pierre Hansen,et al.  Variable Neighborhood Search for Extremal Graphs. 9. Bounding the Irregularity of a Graph , 2001, Graphs and Discovery.

[11]  Pierre Hansen,et al.  Variable Neighborhood Search for Extremal Graphs, 6. Analyzing Bounds for the Connectivity Index , 2001, J. Chem. Inf. Comput. Sci..

[12]  F. K. Bell.  A note on the irregularity of graphs , 1992 .

[13]  Pierre Hansen,et al.  How Far Is, Should and Could Be Conjecture-Making in Graph Theory an Automated Process? , 2001, Graphs and Discovery.

[14]  Jochen Harant,et al.  On the independence number of a graph in terms of order and size , 2001, Discret. Math..

[15]  Ronald D. Dutton,et al.  A compilation of relations between graph invariants - supplement I , 1991, Networks.

[16]  Raimund Seidel,et al.  How Good Are Convex Hull Algorithms? , 1997, Comput. Geom..

[17]  Pierre Hansen,et al.  Variable Neighborhood Search for Extremal Graphs. 10. Comparison of Irregularity Indices for Chemical Trees , 2003, J. Chem. Inf. Model..

[18]  G. Joret,et al.  Turán's theorem and k-connected graphs , 2008 .

[19]  P. Hansen,et al.  Polyenes with maximum HOMO–LUMO gap ☆ , 2001 .

[20]  Pierre Hansen,et al.  Variable neighborhood search for extremal graphs. 5. Three ways to automate finding conjectures , 2000, Discret. Math..

[21]  I. Gitler,et al.  Bounds for graph invariants , 2005 .

[22]  Ronald D. Dutton,et al.  INGRID: A Graph Invariant Manipulator , 1989, J. Symb. Comput..

[23]  G. Ziegler.  Lectures on Polytopes , 1994 .

[24]  P. Hansen,et al.  Variable Neighborhood Search for Extremal Graphs. 12. A Note on the Variance of Bounded Degrees in Graphs , 2004 .

[25]  D. Cvetkovic,et al.  Variable neighborhood search for extremal graphs 3 , 2001 .

[26]  Pierre Hansen,et al.  Variable Neighborhood Search for Extremal Graphs: IV: Chemical Trees with Extremal Connectivity Index , 1998, Comput. Chem..

[27]  Michael O. Albertson,et al.  The Irregularity of a Graph , 1997, Ars Comb..

[28]  Gwenaël Joret,et al.  Turán's theorem and k-connected graphs , 2008, J. Graph Theory.