Maximum Modulus of Independence Roots of Graphs and Trees

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size and its roots are called independence roots . We bound the maximum modulus, maxmod(n) maxmod ( n ) , of an independence root over all graphs on n vertices and the maximum modulus, maxmodT(n) maxmod T ( n ) , of an independence root over all trees on n vertices in terms of n . In particular, we show that log3(maxmod(n))n=13+o(1) log 3 ( maxmod ( n ) ) n = 1 3 + o ( 1 ) and log2(maxmodT(n))n=12+o(1). log 2 ( maxmod T ( n ) ) n = 1 2 + o ( 1 ) .

[1]  Gordon F. Royle,et al.  The Brown-Colbourn conjecture on zeros of reliability polynomials is false , 2004, J. Comb. Theory, Ser. B.

[2]  Johann A. Makowsky,et al.  On the location of roots of graph polynomials , 2013, Eur. J. Comb..

[3]  Bruce E. Sagan,et al.  A Note on Independent Sets in Trees , 1988, SIAM J. Discret. Math..

[4]  M. Plummer Some covering concepts in graphs , 1970 .

[5]  Mohammad Reza Oboudi On the largest real root of independence polynomials of trees , 2018, Ars Comb..

[6]  Jason I. Brown,et al.  On the unimodality of independence polynomials of very well-covered graphs , 2018, Discret. Math..

[7]  Gerard J. Chang,et al.  THE NUMBER OF MAXIMUM INDEPENDENT SETS IN GRAPHS , 2000 .

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

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

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

[11]  Jason I. Brown,et al.  Bounding the Roots of Independence Polynomials , 2001, Ars Comb..

[12]  Odile Favaron Very well covered graphs , 1982, Discret. Math..

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

[14]  Charles J. Colbourn,et al.  Roots of the Reliability Polynomial , 1992, SIAM J. Discret. Math..

[15]  Brendan D. McKay,et al.  Practical graph isomorphism, II , 2013, J. Symb. Comput..

[16]  J. Moon,et al.  On cliques in graphs , 1965 .

[17]  Péter Csikvári Note on the Smallest Root of the Independence Polynomial , 2013, Comb. Probab. Comput..

[18]  Jason Brown,et al.  On the roots of all-terminal reliability polynomials , 2017, Discret. Math..

[19]  Herbert S. Wilf,et al.  The number of maximal independent sets in a tree , 1986 .

[20]  Neil Hindman,et al.  Additive and Multiplicative Ramsey Theorems in N-Some Elementary Results , 1993, Comb. Probab. Comput..

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

[22]  Jason I. Brown,et al.  On the Stability of Independence Polynomials , 2018, Electron. J. Comb..

[23]  Richard J. Nowakowski,et al.  A Characterization of Well Covered Graphs of Girth 5 or Greater , 1993, J. Comb. Theory, Ser. B.

[24]  S. Kakeya,et al.  On the Limits of the Roots of an Algebraic Equation with Positive Coefficients , 1912 .

[25]  Jason I. Brown,et al.  Roots of Independence Polynomials of Well Covered Graphs , 2000 .

[26]  Paul D. Seymour,et al.  The roots of the independence polynomial of a clawfree graph , 2007, J. Comb. Theory B.

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

[28]  Vadim E. Levit,et al.  On the roots of independence polynomials of almost all very well-covered graphs , 2008, Discret. Appl. Math..