Distinguishing graphs by their left and right homomorphism profiles

We introduce a new property of graphs called 'q-state Potts uniqueness' and relate it to chromatic and Tutte uniqueness, and also to 'chromatic-flow uniqueness', recently studied by Duan, Wu and Yu. We establish for which edge-weighted graphs H homomorphism functions from multigraphs G to H are specializations of the Tutte polynomial of G, in particular answering a question of Freedman, Lovasz and Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from multigraphs G to H are specializations of the 'edge elimination polynomial' of Averbouch, Godlin and Makowsky and the 'induced subgraph polynomial' of Tittmann, Averbouch and Makowsky. Unifying the study of these and related problems is the notion of the left and right homomorphism profiles of a graph.

[1]  Johann A. Makowsky,et al.  The enumeration of vertex induced subgraphs with respect to the number of components , 2008, Eur. J. Comb..

[2]  Jaroslav Nesetril,et al.  Graphs and homomorphisms , 2004, Oxford lecture series in mathematics and its applications.

[3]  W. T. Tutte.  A Class Of Abelian Groups , 1956, Canadian Journal of Mathematics.

[4]  L. Lovasz,et al.  Reflection positivity, rank connectivity, and homomorphism of graphs , 2004, math/0404468.

[5]  Jaroslav Nesetril,et al.  Graph homomorphisms, the Tutte polynomial and "q-state Potts uniqueness" , 2009, Electron. Notes Discret. Math..

[6]  Delia Garijo,et al.  Tutte Uniqueness of Locally Grid Graphs , 2009, Ars Comb..

[7]  Norman Biggs.  Algebraic Graph Theory: Index , 1974 .

[8]  Peter Tittmann,et al.  A new two-variable generalization of the chromatic polynomial , 2003, Discret. Math. Theor. Comput. Sci..

[9]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[10]  Marc Noy,et al.  Locally grid graphs: classification and Tutte uniqueness , 2003, Discret. Math..

[11]  Ronald C. Read,et al.  A note on chain lengths and the Tutte polynomial , 2008, Discret. Math..

[12]  Noga Alon,et al.  Colorings and orientations of graphs , 1992, Comb..

[13]  Johann A. Makowsky,et al.  An extension of the bivariate chromatic polynomial , 2010, Eur. J. Comb..

[14]  A. Goodall,et al.  Fourier analysis on finite Abelian groups: some graphical applications , 2006 .

[15]  Martin E. Dyer,et al.  The complexity of counting graph homomorphisms , 2000, Random Struct. Algorithms.

[16]  Zdenek Dvorak,et al.  On recognizing graphs by numbers of homomorphisms , 2010, J. Graph Theory.

[17]  L. Lovász.  Operations with structures , 1967 .

[18]  Y. H. Peng,et al.  Chromatic uniqueness of certain complete tripartite graphs , 2009, Ars Comb..

[19]  Marc Noy,et al.  On Graphs Determined by Their Tutte Polynomials , 2004, Graphs Comb..

[20]  John S. Rowlinson,et al.  New Model for the Study of Liquid–Vapor Phase Transitions , 1970 .

[21]  Kee L. Teo,et al.  The search for chromatically unique graphs , 1990, Graphs Comb..

[22]  Jaroslav Nesetril,et al.  Homomorphisms and polynomial invariants of graphs , 2009, Eur. J. Comb..

[23]  Alexander Schrijver,et al.  Dual graph homomorphism functions , 2010, J. Comb. Theory, Ser. A.

[24]  Kee L. Teo,et al.  The search for chromatically unique graphs - II , 1997, Discret. Math..

[25]  Klas Markström,et al.  The bivariate Ising polynomial of a graph , 2009, Discret. Appl. Math..

[26]  Tom Brylawski,et al.  Matroid Applications: The Tutte Polynomial and Its Applications , 1992 .

[27]  Qinglin Yu,et al.  On chromatic and flow polynomial unique graphs , 2008, Discret. Appl. Math..

[28]  V. Sós,et al.  Counting Graph Homomorphisms , 2006 .

[29]  F. Harary.  New directions in the theory of graphs , 1973 .

[30]  Béla Bollobás,et al.  Modern Graph Theory , 2002, Graduate Texts in Mathematics.

[31]  Béla Bollobás,et al.  Contraction-Deletion Invariants for Graphs , 2000, J. Comb. Theory B.

[32]  László Lovász,et al.  The rank of connection matrices and the dimension of graph algebras , 2004, Eur. J. Comb..

[33]  P. Valtr,et al.  Topics in Discrete Mathematics , 2006 .

[34]  W. Haemers,et al.  Which graphs are determined by their spectrum , 2003 .

[35]  Béla Bollobás,et al.  A Tutte Polynomial for Coloured Graphs , 1999, Combinatorics, Probability and Computing.

[36]  Marc Noy,et al.  Graphs determined by polynomial invariants , 2003, Theor. Comput. Sci..