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A note on flow polynomials of graphs

2009
*
Discrete Mathematics
*

Using the decomposition theory

doi:10.1016/j.disc.2008.01.050
fatcat:vizrl32mpbdrdmg6xxjhtpgkiu
*of*modular and integral*flow**polynomials*, we answer*a*problem*of*Beck and Zaslavsky, by providing*a*general situation in which the integral*flow**polynomial*is*a*multiple ...*of*the modular*flow**polynomial*. ... The second author was also supported by the 973 Project*on*Mathematical Mechanization, the PCSIRT Project*of*the Ministry*of*Education, the Ministry*of*Science and Technology, and the National Science ...##
###
Chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs

2016
*
Discrete Applied Mathematics
*

*A*generalized vertex join

*of*

*a*

*graph*is obtained by joining an arbitrary multiset

*of*its vertices to

*a*new vertex. ... We present

*a*low-order

*polynomial*time algorithm for computing the chromatic

*polynomials*

*of*generalized vertex joins

*of*trees; by duality, this algorithm can also be used to compute the

*flow*

*polynomials*...

*Note*that (13) depends only

*on*the sequence

*of*face-sizes

*of*C S and hence only

*on*S. Finally, to find the

*flow*

*polynomial*

*of*C S ,

*note*that by (4) , F (C S ) = 1 t P (C * S ). ...

##
###
Zeros of Chromatic and Flow Polynomials of Graphs
[article]

2002
*
arXiv
*
pre-print

We survey results and conjectures concerning the zero distribution

arXiv:math/0205047v2
fatcat:kilv3jtqvraejfzvjnstbprn3a
*of*chromatic and*flow**polynomials**of**graphs*, and characteristic*polynomials**of*matroids. ... Acknowledgements I would like to thank Alan Sokal and Douglas Woodall for many helpful remarks and inspiring conversations*on*chromatic roots which have greatly contributed to this survey. ... Characteristic*polynomials**of*matroids provide*a*common generalization*of*chromatic and*flow**polynomials**of**graphs*. ...##
###
Decomposition of the Flow Polynomial

1997
*
Graphs and Combinatorics
*

The

doi:10.1007/bf03352995
fatcat:dx4zx4x6pza4zat32lr6f4m74a
*flow**polynomials*denote the number*of*nowhere-zero*flows**on**graphs*, and are related to the well-known Tutte*polynomials*and chromatic*polynomials*. ... Another application*of*the decomposition results is that if*a*bridgeless*graph*G does not admit*a*nowhere-zero k-*flow*and G has*a*small vertex-or edge-cut, then*a*proper bridgeless subgraph*of*G (*a**graph*... The first author would like to thank Hiroshi Imai who read through the earlier draft*of*this paper and gave her helpful comments . ...##
###
Nowhere-Zero k⃗-Flows on Graphs
[article]

2013
*
arXiv
*
pre-print

We introduce and study

arXiv:1305.2456v1
fatcat:vcbhekxkdvdijln3wmuqkonwwi
*a*multivariate function that counts nowhere-zero*flows**on**a**graph*G, in which each edge*of*G has an individual capacity. ... We prove that the associated counting function is*a*piecewise-defined*polynomial*in these capacities, which satisfy*a*combinatorial reciprocity law that incorporates totally cyclic orientations*of*G. ... For*a*bridgeless*graph*G, the multivariate*flow*counting function ϕ G (k) is*a*piecewisedefined*polynomial**of*degree ξ G . ...##
###
Chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs
[article]

2015
*
arXiv
*
pre-print

We present

arXiv:1501.04388v1
fatcat:owu5rz5vlbasrooos7kqz2nyf4
*a*low-order*polynomial*time algorithm for finding the chromatic*polynomials**of*generalized vertex joins*of*trees, and by duality we find the*flow**polynomials**of*arbitrary outerplanar*graphs*. ...*A*generalized vertex join*of**a**graph*is obtained by joining an arbitrary multiset*of*its vertices to*a*new vertex. ... [2, 17]*on*chromatic*polynomials**of*outerplanar*graphs*and*flow**polynomials**of*their duals, by characterizing the*flow**polynomials**of*outerplanar*graphs*and the chromatic*polynomials**of*their duals. ...##
###
A note on chain lengths and the Tutte polynomial

2008
*
Discrete Mathematics
*

We show that the number

doi:10.1016/j.disc.2006.09.049
fatcat:zcumh3lbpbbclnoct7p6n67uvq
*of*chains*of*given length in*a**graph*G can be easily found from the Tutte*polynomial**of*G. Hence two Tutte-equivalent*graphs*will have the same distribution*of*chain lengths. ... We give two applications*of*this latter statement. We also give the dual results for the numbers*of*multiple edges with given muliplicities. ... We deduce that all s-theta-*graphs*are Tutte-unique. (b) Consider any homeomorph, G,*of*K 4 -the complete*graph**on*four vertices. ...##
###
A survey on the study of real zeros of flow polynomials

2019
*
Journal of Graph Theory
*

For

doi:10.1002/jgt.22458
fatcat:j4jrgxgvrjf6hc5kcgpwkp2odq
*a*bridgeless*graph*G, its*flow**polynomial*is defined to be the function F(G,q) which counts the number*of*nonwhere-zero Γ-*flows**on*an orientation*of*G whenever q is*a*positive integer and Γ is an additive ... This article gives*a*survey*on*the results and problems*on*the study*of*real zeros*of**flow**polynomials*. ... The zeros*of*the*flow**polynomial*F (G, q) are called the*flow*roots*of*G. This article focuses*on*giving*a*review*on*the study*of*real*flow*roots*of**graphs*. ...##
###
On chromatic and flow polynomial unique graphs

2008
*
Discrete Applied Mathematics
*

It is known that the chromatic

doi:10.1016/j.dam.2007.10.010
fatcat:6fxjxx7owfh73orj6su54oih7m
*polynomial*and*flow**polynomial**of**a**graph*are two important evaluations*of*its Tutte*polynomial*, both*of*which contain much information*of*the*graph*. ... Noy,*On**graphs*determined by their Tutte*polynomial*,*Graphs*Comb. 20 (2004) 105-119] that these classes*of**graphs*are Tutte*polynomial*unique. ... Acknowledgments The authors wish to thank the referees for providing many helpful comments*on*an earlier version*of*this paper. ...##
###
Tutte relations, TQFT, and planarity of cubic graphs
[article]

2015
*
arXiv
*
pre-print

*A*version

*of*the Tutte linear relation for the

*flow*

*polynomial*at (3-√(5))/2 is shown to give

*a*planarity criterion for 3-connected cubic

*graphs*. ...

*A*conjecture is formulated that the golden identity for the

*flow*

*polynomial*characterizes planarity

*of*cubic

*graphs*as well. ... Ian Agol was partially supported by NSF grant DMS-1406301, and by

*a*Simons Investigator grant. Vyacheslav Krushkal was partially supported by NSF grant DMS-1309178. ...

##
###
Non-interfering network flows
[chapter]

1992
*
Lecture Notes in Computer Science
*

*A*

*polynomial*time algorithm is outlined for arbitrary d when the underlying network is planar and how a.n integral

*flow*ca.n be obtained from

*a*. fractional

*one*. ... We consider

*a*generalization

*of*the maximum

*flow*problem where instead

*of*bounding the amount

*of*

*flow*which passes through an arc, we bound the amount

*of*

*flow*passing "near" an arc. ... This paper stemmed from discussions at

*a*workshop at Belairs Research Institute. We thank Wayne Hunt and his colleagues for their hospitality and providing an ideal working environment. ...

##
###
Chromatic polynomials of homeomorphism classes of graphs

1999
*
Discrete Mathematics
*

We study

doi:10.1016/s0012-365x(98)00378-1
fatcat:qp4tcsbp2zalhpogp4rha3nq6e
*a*multilinear*polynomial*which subsumes the chromatic*polynomials**of*all the*graphs*in*a*given homeomorphism class. ... We show that this*polynomial*can be extended to include further families*of*homeomorphic*graphs*, and derive some properties*of*its coefficients. ... M is then the*graph*in Fig. 15 for which the*flow**polynomial*is (-co)'*oF*(M'). ...##
###
Some polynomials of flower graphs

2007
*
International Mathematical Forum
*

We define

doi:10.12988/imf.2007.07221
fatcat:jabdw4nibnbh3pjx6xvtje3hyq
*a*class*of**graphs*called flower and give some properties*of*these*graphs*. Then the explicit expressions*of*the chromatic*polynomial*and the*flow**polynomial*is given. ... Further, we give classes*of**graphs*with the same chromatic and*flow**polynomials*. Mathematics Subject Classification: 05C99 ... It should be*noted*that the explicit expression*of*the*flow**polynomial*given, is only for flower*graphs*which are not complete. ...##
###
Structure of the flow and Yamada polynomials of cubic graphs
[article]

2018
*
arXiv
*
pre-print

An application is given to the structure

arXiv:1801.00502v1
fatcat:x7xdh2brfbfghlfd5zsfrgwcbi
*of*the*flow**polynomial**of*cubic*graphs*at zero. ... The golden identity for the*flow**polynomial*is conjectured to characterize planarity*of*cubic*graphs*, and we prove this conjecture for*a*certain infinite family*of*non-planar*graphs*. ... We would like to thank Gordon Royle for many discussions, and also for sharing with us numerical data*on**graph**polynomials*. We also thank Kyle Miller for helpful comments. V. ...##
###
From generalized permutahedra to Grothendieck polynomials via flow polytopes

2020
*
Algebraic Combinatorics
*

To each dissection

doi:10.5802/alco.136
fatcat:q2c56txehjgedgm66zh3g5myly
*of**a**flow*polytope, we associate*a**polynomial*, called the left-degree*polynomial*, which we show is invariant*of*the dissection considered (proven independently by Grinberg). ... We study*a*family*of*dissections*of**flow*polytopes arising from the subdivision algebra. ... We are grateful to the anonymous referees for their helpful and detailed feedback, which improved the exposition*of*the paper. ...
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