Academia.eduAcademia.edu

Outline

Title
Abstract
Key Takeaways
Introduction
Conclusions and Open Problems
References
All Topics
Mathematics
Pure Mathematics

On the Location of Roots of Graph Polynomials

Profile image of Elena RavveElena Ravve

2013, Electronic Notes in Discrete Mathematics

https://doi.org/10.1016/J.ENDM.2013.07.033
visibility

…

description

21 pages

link

1 file

Abstract

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.

Key takeaways
sparkles

AI

  1. The paper explores how roots of graph polynomials reflect graph properties.
  2. It establishes d.p.-equivalence and prefactor equivalence for graph polynomials.
  3. The authors demonstrate modification theorems for root locations under equivalences.
  4. Real roots of graph polynomials can be made dense or bounded through transformations.
  5. The findings suggest that semantic significance in graph polynomials is a research challenge.

Related papers

The higher-order matching polynomial of a graph
Mario Estrada

International Journal of Mathematics and Mathematical Sciences, 2005

View PDFchevron_right
On the chromatic polynomial of a graph
Paolo Nobili

Mathematical Programming, 2002

Let P(G, λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ ≥ n,

View PDFchevron_right
The Characteristic Polynomials of Symmetric Graphs
Nafaa Chbili

Symmetry, 2018

In this paper, we study the way the symmetries of a given graph are reflected in its characteristic polynomials. Our aim is not only to find obstructions for graph symmetries in terms of its polynomials but also to measure how faithful these algebraic invariants are with respect to symmetry. Let p be an odd prime and Γ be a finite graph whose automorphism group contains an element h of order p. Assume that the finite cyclic group generated by h acts semi-freely on the set of vertices of Γ with fixed set F. We prove that the characteristic polynomial of Γ, with coefficients in the finite field of p elements, is the product of the characteristic polynomial of the induced subgraph Γ[F] by one of Γ \ F. A similar congruence holds for the characteristic polynomial of the Laplacian matrix of Γ.

View PDFchevron_right
More connections between the matching polynomial and the chromatic polynomial
Marcelino Ramírez

AKCE International Journal of Graphs and Combinatorics, 2019

The connection between the matching polynomial and the chromatic polynomial for triangle-free graphs was revealed in the work of Farrell and Whitehead. We extend this result to all graph by mirroring the corresponding result of Godsil and Gutman for the acyclic polynomial and the characteristic polynomial. We also reintroduce the clique polynomial of Farrell as an evaluation of the U-polynomial of Noble and Welsh, which also happens to contain the matching and chromatic polynomials. c

View PDFchevron_right
Comparative Theory for Graph Polynomials (Dagstuhl Seminar 19401)
Andrew Goodall

Dagstuhl Reports, 2019

This report documents the programme and outcomes of Dagstuhl Seminar 19401 "Comparative Theory for Graph Polynomials". The study of graph polynomials has become increasingly active, with new applications and new graph polynomials being discovered each year. The genera of graph polynomials are diverse, and their interconnections are rich. Experts in the field are finding that proof techniques and results established in one area can be successfully extended to others. From this a general theory is emerging that encapsulates the deeper interconnections between families of graph polynomials and the various techniques, computational approaches, and methodologies applied to them. The overarching aim of this Seminar was to exploit commonalities among polynomial invariants of graphs, matroids, and related combinatorial structures. Model-theoretic, computational and other methods were used in order to initiate a comparative theory that collects the current state of knowledge into a more cohesive and powerful framework.

View PDFchevron_right
Characteristic polynomials of some algebraic graphs
M R Jadeja

2020

The zero divisor graph Γ(R) of a commutative ring R is a graph whose vertices are non-zero zero divisors of R and two vertices are adjacent if their product is zero. The characteristic polynomial of matrix M is defined as |λ I−M| and roots of the characteristic polynomial are known as eigenvalues of M. We investigate eigenvalues and characteristic polynomials for some zero divisor graphs.

View PDFchevron_right
Graph polynomials and symmetries
Nafaa Chbili

Journal of Algebra and Its Applications, 2018

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.

View PDFchevron_right
A characteristic polynomial for rooted graphs and rooted digraphs
Liz McMahon

Discrete Mathematics, 2001

We consider the one-variable characteristic polynomial p(G;) in two settings. When G is a rooted digraph, we show that this polynomial essentially counts the number of sinks in G. When G is a rooted graph, we give combinatorial interpretations of several coe cients and the degree of p(G;). In particular, |p(G; 0)| is the number of acyclic orientations of G, while the degree of p(G;) gives the size of the minimum tree cover (every edge of G is adjacent to some edge of T), and the leading coe cient gives the number of such covers. Finally, we consider the class of rooted fans in detail; here p(G;) shows cyclotomic behavior.

View PDFchevron_right
Chromatic Polynomials and Chromaticity of Graphs
Fengming Dong

2005

View PDFchevron_right
On the Roots of the Modified Orbit Polynomial of a Graph
Modjtaba Ghorbani

Symmetry, 2021

The definition of orbit polynomial is based on the size of orbits of a graph which is OG(x)=∑ix|Oi|, where O1,…,Ok are all orbits of graph G. It is a well-known fact that according to Descartes’ rule of signs, the new polynomial 1−OG(x) has a positive root in (0,1), which is unique and it is a relevant measure of the symmetry of a graph. In the current work, several bounds for the unique and positive zero of modified orbit polynomial 1−OG(x) are investigated. Besides, the relation between the unique positive root of OG in terms of the structure of G is presented.

View PDFchevron_right

References (44)

  1. M. Aigner. A course in enumeration. Graduate Texts in Mathematics. Springer, 2007.
  2. S. Akbari, S. Alikhani, and Y.-H. Peng. Characterization of graphs using domination poly- nomials. Eur. J. Comb., 31(7):1714-1724, 2010.
  3. S. Akbari and M. R. Oboudi. On the edge cover polynomial of a graph. Eur. J. Comb., 34(2):297-321, 2013.
  4. S. Alikhani. Dominating Sets and Domination Polynomials of Graphs. PhD thesis, Universiti Putra Malaysia, 2009. http://psasir.upm.edu.my/7250/.
  5. J. L. Arocha and B. Llano. Mean value for the matching and dominating polynomial. Dis- cussiones Mathematicae Graph Theory, 20(1):57-69, 2000.
  6. I. Averbouch. Completeness and Universality Properties of Graph Invariants and Graph Polynomials. PhD thesis, Technion -Israel Institute of Technology, Haifa, Israel, 2011.
  7. I. Averbouch, B. Godlin, and J.A. Makowsky. An extension of the bivariate chromatic poly- nomial. European Journal of Combinatorics, 31(1):1-17, 2010.
  8. A. T. Balaban. Solved and unsolved problems in chemical graph theory. quo vadis, graph theory? Ann. Discrete Math., 35:109-126, 1993.
  9. A. T. Balaban. Chemical graphs: Looking back and glimpsing ahead. Journal of Chemical Information and Computer Science, 35:339-350, 1995.
  10. B. Bollobás. Modern Graph Theory. Springer, 1998.
  11. F. Brenti, G.F. Royle, and D.G. Wagner. Location of zeros of chromatic and related polyno- mials of graphs. Canadian Journal of Mathematics, 46:55-80, 1994.
  12. A.E. Brouwer and W.H. Haemers. Spectra of Graphs. Springer Universitext. Springer, 2012.
  13. J.I. Brown, C.A. Hickman, and R.J. Nowakowski. On the location of roots of independence polynomials. Journal of Algebraic Combinatorics, 19.3:273-282, 2004.
  14. J.I. Brown and J. Tufts. On the roots of domination polynomials. Graphs and Combinatorics, published online:DOI 10.1007/s00373-013-1306-z, 2013.
  15. P. Csikvári and M. R. Oboudi. On the roots of edge cover polynomials of graphs. Eur. J. Comb., 32(8):1407-1416, 2011.
  16. D.M. Cvetković, M. Doob, and H. Sachs. Spectra of Graphs. Johann Ambrosius Barth, 3rd edition, 1995.
  17. P. de la Harpe and F. Jaeger. Chromatic invariants for finite graphs: Theme and polynomial variations. Linear Algebra and its Applications, 226-228:687-722, 1995.
  18. F.M. Dong, M.D. Hendy, K.L. Teo, and C.H.C. Little. The vertex-cover polynomial of a graph. Discrete Mathematics, 250:71-78, 2002.
  19. F.M. Dong, K.M. Koh, and K.L. Teo. Chromatic Polynomials and Chromaticity of Graphs. World Scientific, 2005.
  20. J. Ellis-Monaghan and C. Merino. Graph polynomials and their applications i: The Tutte polynomial. arXiv 0803.3079v2 [math.CO], 2008.
  21. J. Ellis-Monaghan and C. Merino. Graph polynomials and their applications ii: Interrelations and interpretations. arXiv 0806.4699v1 [math.CO], 2008.
  22. J. A. Ellis-Monaghan and I. Sarmiento. A recipe theorem for the topological Tutte polynomial of Bollobás and Riordan. Eur. J. Comb., 32(6):782-794, 2011.
  23. D. Garijo, A. Goodall, and J. Nešetřil. Polynomial graph invariants from homomorphism numbers. arXiv:1308.3999 [math.CO], 2013.
  24. B. Godlin, E. Katz, and J.A. Makowsky. Graph polynomials: From recursive definitions to subset expansion formulas. Journal of Logic and Computation, 22(2):237-265, 2012.
  25. M. Goldwurm and M. Santini. Clique polynomials have a unique root of smallest modulus. Inf. Process. Lett., 75(3):127-132, 2000.
  26. I. Gutman and F. Harary. Generalizations of the matching polynomial. Utilitas Mathematicae, 24:97-106, 1983.
  27. C.J. Heilmann and E.H. Lieb. Theory of monomer-dymer systems. Comm. Math. Phys, 25:190-232, 1972.
  28. P. Henrici. Applied and Computational Complex Analysis, volume 1. Wiley Classics Library. John Wiley, 1988.
  29. C. Hoede and X. Li. Clique polynomials and independent set polynomials of graphs. Discrete Mathematics, 125:219-228, 1994.
  30. R. Hoshino. Independence Polynomials of Circulant Graphs. PhD thesis, Dalhousie Univer- sity, Halifax, Nova Scotia, 2007.
  31. T. Kotek, J.A. Makowsky, and B. Zilber. On counting generalized colorings. In M. Grohe and J.A. Makowsky, editors, Model Theoretic Methods in Finite Combinatorics, volume 558 of Contemporary Mathematics, pages 207-242. American Mathematical Society, 2011.
  32. T. Kotek, J. Preen, F. Simon, P. Tittmann, and M. Trinks. Recurrence relations and splitting formulas for the domination polynomial. Electr. J. Comb., 19(3):P47, 2012.
  33. V.E. Levit and E. Mandrescu. The independence polynomial of a graph -a survey. In S. Boza- palidis, A. Kalampakas, and G. Rahonis, editors, Proceedings of the 1st International Con- ference on Algebraic Informatics, pages 233-254. Aristotle University of Thessaloniki, De- partment of Mathematics, Thessaloniki, 2005.
  34. L.Lovász. Large Networks and Graph Limits, volume 60 of Colloquium Publications. AMS, 2012.
  35. L. Lovász and M.D. Plummer. Matching Theory, volume 29 of Annals of Discrete Mathe- matics. North Holland, 1986.
  36. J.A. Makowsky. Algorithmic uses of the Feferman-Vaught theorem. Annals of Pure and Applied Logic, 126.1-3:159-213, 2004.
  37. J.A. Makowsky and E.V. Ravve. Logical methods in combinatorics, Lecture 11. Course given in 2009 under the number 236605, Advanced Topics, available at: http://www.cs.technion.ac.il/ janos/COURSES/236605-09/lec11.pdf.
  38. J.A. Makowsky and E.V. Ravve. On the location of roots of graph polynomials. Electronic Notes in Discrete Mathematics, 43:201-206, 2013.
  39. C. Merino and S. D. Noble. The equivalence of two graph polynomials and a symmetric function. Combinatorics, Probability & Computing, 18(4):601-615, 2009.
  40. A. D. Sokal. Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions. Combinatorics, Probability & Computing, 10(1):41-77, 2001.
  41. A. D. Sokal. Chromatic roots are dense in the whole complex plane. Combinatorics, Proba- bility & Computing, 13(2):221-261, 2004.
  42. A. D. Sokal. The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In Survey in Combinatorics, 2005, volume 327 of London Mathematical Society Lecture Notes, pages 173-226, 2005.
  43. P. Tittmann, I. Averbouch, and J.A. Makowsky. The enumeration of vertex induced sub- graphs with respect to the number of components. European Journal of Combinatorics, 32(7):954-974, 2011.
  44. N. Trinajstić. Chemical Graph Theory. CRC Press, 2nd edition, 1992. Computer Science Department, Technion-IIT, Haifa, Israel E-mail address: janos@.cs.technion.ac.il

Related papers

The characteristic polynomial of a graph
Abbe Mowshowitz

Journal of Combinatorial Theory, Series B, 1972

The present paper is addressed to the problem of determining under what conditions the characteristic polynomial of the adjacency matrix of a graph distinguishes between non-isomorphic graphs. A formula for the coeiiicients of the characteristic polynomial of an arbitrary digraph is derived, and the polynomial of a tree is examined in depth. It is shown that the coefFicients of the polynomial of a tree count matchings. Several recurrence relations are also given for computing the coefficients. An appendix is provided which lists n-node trees (2 < N < 10) together with the coefficients of their polynomials. It should be aoted that this list corrects some errors in the earlier table of [I].

View PDFchevron_right
Basics on Chromatic Polynomials in Graphs
IJCRM Journal

IJCRM.COM, 2016

This article is a general introduction of chromatic polynomials.In this,Chromatic polynomials are defined,their connection between the theory of chromatic polynomials and coloring of graphs.Also it explains the chromatic polynomials of total graphs .It gives the basic concepts of chromatic polynomials in graph theory.

View PDFchevron_right
Polynomials in graph theory
Alexey Bogatov

2015

In this paper we observe the problem of counting graph colorings using polynomials. Several reformulations of The Four Color Conjecture are considered (among them algebraic, probabilistic and arithmetic). In the last section Tutte polynomials are mentioned. 1.

View PDFchevron_right
Matching number and characteristic polynomial of a graph
Musa demirci

Journal of Taibah University for Science

Matching number and the spectral properties depending on the characteristic polynomial of a graph obtained by means of the adjacency polynomial has many interesting applications in different areas of science. There are some work giving the relation of these two areas. Here the relations between these two notions are considered and several general results giving this relations are obtained. A result given for only unicyclic graphs is generalized. There are some methods for determining the matching number of a graph in literature. Usually nullity, spanning trees and several graph parts are used to do this. Here, as a new method, the conditions for calculating the matching number of a graph by means of the coefficients of the characteristic polynomial of the graph are determined. Finally some results on the matching number of graphs are obtained.

View PDFchevron_right
Graphs which their certain polynomials have few distinct roots-a survey
Saeid Alikhani

2011

Let G = (V, E) be a simple graph. We consider domination polynomial, matching polynomial and edge cover polynomial of G. Graphs which their polynomials have few roots can give sometimes a very surprising information about the structure of the graph. In this paper we study graphs which their domination polynomial, independence polynomial and edge cover polynomial have few roots.

View PDFchevron_right

Related topics

  • Pure Mathematics
    • 580 California St., Suite 400
    San Francisco, CA, 94104
    © 2025 Academia. All rights reserved