On the location of roots of graph polynomials

Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph.

[1]  Feng Ming Dong,et al.  The vertex-cover polynomial of a graph , 2002, Discret. Math..

[2]  J. Helton,et al.  Linear matrix inequality representation of sets , 2003, math/0306180.

[3]  Alan D. Sokal.  The multivariate Tutte polynomial (alias Potts model) for graphs and matroids , 2005, Surveys in Combinatorics.

[4]  Criel Merino,et al.  Graph Polynomials and Their Applications II: Interrelations and Interpretations , 2008, Structural Analysis of Complex Networks.

[5]  Criel Merino,et al.  Graph Polynomials and Their Applications I: The Tutte Polynomial , 2008, Structural Analysis of Complex Networks.

[6]  Petter Brändén,et al.  Polynomials with the half-plane property and matroid theory , 2007 .

[7]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

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

[9]  Tomer Kotek,et al.  Recurrence Relations and Splitting Formulas for the Domination Polynomial , 2012, Electron. J. Comb..

[10]  K. Koh,et al.  Chromatic polynomials and chro-maticity of graphs , 2005 .

[11]  G. Royle,et al.  Location of Zeros of Chromatic and Related Polynomials of Graphs , 1994, Canadian Journal of Mathematics.

[12]  Alexandru T. Balaban,et al.  Chemical Graphs: Looking Back and Glimpsing Ahead , 1995, J. Chem. Inf. Comput. Sci..

[13]  Alexandru T. Balaban,et al.  Solved and Unsolved Problems in Chemical Graph Theory , 1993 .

[14]  Samuel Kolins.  Topological Methods in Combinatorics , 2010 .

[15]  O. J. Heilmann,et al.  Theory of monomer-dimer systems , 1972 .

[16]  O. J. Heilmann,et al.  Theory of monomer-dimer systems , 1972 .

[17]  Vadim E. Levit,et al.  The independence polynomial of a graph - a survey , 2005 .

[18]  Bernardo Llano,et al.  Mean value for the matching and dominating polynomial , 2000, Discuss. Math. Graph Theory.

[19]  David G. Wagner,et al.  A criterion for the half-plane property , 2007, Discret. Math..

[20]  Criel Merino,et al.  The Equivalence of Two Graph Polynomials and a Symmetric Function , 2008, Combinatorics, Probability and Computing.

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

[22]  Xueliang Li,et al.  Clique polynomials and independent set polynomials of graphs , 1994, Discret. Math..

[23]  Ilia Averbouch.  Completeness and universality properties of graph invariants and graph polynomials , 2011 .

[24]  Joanna A. Ellis-Monaghan,et al.  A recipe theorem for the topological Tutte polynomial of Bollobás and Riordan , 2009, Eur. J. Comb..

[25]  Jason I. Brown,et al.  On the Roots of Domination Polynomials , 2014, Graphs Comb..

[26]  Saeid Alikhani,et al.  Dominating sets and Domination polynomials of Cycles , 2009, 0905.3268.

[27]  Jason I. Brown,et al.  On the Location of Roots of Independence Polynomials , 2004 .

[28]  François Jaeger,et al.  Chromatic invariants for finite graphs: Theme and polynomial variations , 1995 .

[29]  David G. Wagner,et al.  Homogeneous multivariate polynomials with the half-plane property , 2004, Adv. Appl. Math..

[30]  Alan D. Sokal,et al.  Chromatic Roots are Dense in the Whole Complex Plane , 2000, Combinatorics, Probability and Computing.

[31]  Saieed Akbari,et al.  Characterization of graphs using domination polynomials , 2010, Eur. J. Comb..

[32]  Johann A. Makowsky,et al.  Graph Polynomials: From Recursive Definitions to Subset Expansion Formulas , 2008, J. Log. Comput..

[33]  J. Borcea,et al.  Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products , 2006, math/0607755.

[34]  P. Brand'en.  Polynomials with the half-plane property and matroid theory , 2006, math/0605678.

[35]  Péter Csikvári,et al.  On the roots of edge cover polynomials of graphs , 2011, Eur. J. Comb..

[36]  D. Wagner,et al.  Multivariate stable polynomials: theory and applications , 2009, 0911.3569.

[37]  Massimo Santini,et al.  Clique polynomials have a unique root of smallest modulus , 2000, Inf. Process. Lett..

[38]  Michael Doob,et al.  Spectra of graphs , 1980 .

[39]  Johann A. Makowsky,et al.  On Counting Generalized Colorings , 2008, CSL.

[40]  Richard Hoshino,et al.  INDEPENDENCE POLYNOMIALS OF CIRCULANT GRAPHS , 2007 .

[41]  Johann A. Makowsky,et al.  Algorithmic uses of the Feferman-Vaught Theorem , 2004, Ann. Pure Appl. Log..

[42]  Martin Aigner,et al.  A Course in Enumeration , 2007 .

[43]  Saieed Akbari,et al.  On the edge cover polynomial of a graph , 2013, Eur. J. Comb..

[44]  K. Murasugi,et al.  Various stabilities of the Alexander polynomials of knots and links , 2013, 1307.1578.

[45]  Willem H. Haemers,et al.  Spectra of Graphs , 2011 .

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

[47]  Alan D. Sokal,et al.  Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions , 1999, Combinatorics, Probability and Computing.

[48]  Saeid Alikhani,et al.  Dominating sets and domination polynomials of certain graphs, II , 2010 .

[49]  Johann A. Makowsky,et al.  On the Location of Roots of Graph Polynomials , 2013, Electron. Notes Discret. Math..

[50]  N. Trinajstic.  Chemical Graph Theory , 1992 .

[51]  Jaroslav Nesetril,et al.  Polynomial graph invariants from homomorphism numbers , 2013, Discret. Math..