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A note on flow polynomials of graphs

Beifang Chen, Arthur L.B. Yang
2009 Discrete Mathematics  
Using the decomposition theory 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  ... 
doi:10.1016/j.disc.2008.01.050 fatcat:vizrl32mpbdrdmg6xxjhtpgkiu

Chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs

Boris Brimkov, Illya V. Hicks
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 ).  ... 
doi:10.1016/j.dam.2015.10.016 fatcat:ceyvgshtw5g7ncj75j7ddkw5ay

Zeros of Chromatic and Flow Polynomials of Graphs [article]

Bill Jackson
2002 arXiv   pre-print
We survey results and conjectures concerning the zero distribution 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.  ... 
arXiv:math/0205047v2 fatcat:kilv3jtqvraejfzvjnstbprn3a

Decomposition of the Flow Polynomial

K. Sekine, C. Q. Zhang
1997 Graphs and Combinatorics  
The 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 .  ... 
doi:10.1007/bf03352995 fatcat:dx4zx4x6pza4zat32lr6f4m74a

Nowhere-Zero k⃗-Flows on Graphs [article]

Matthias Beck, Alyssa Cuyjet, Gordon Rojas Kirby, Molly Stubblefield,, Michael Young
2013 arXiv   pre-print
We introduce and study 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 .  ... 
arXiv:1305.2456v1 fatcat:vcbhekxkdvdijln3wmuqkonwwi

Chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs [article]

Boris Brimkov, Illya V. Hicks
2015 arXiv   pre-print
We present 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.  ... 
arXiv:1501.04388v1 fatcat:owu5rz5vlbasrooos7kqz2nyf4

A note on chain lengths and the Tutte polynomial

Ronald C. Read, Earl Glen Whitehead Jr.
2008 Discrete Mathematics  
We show that the number 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.  ... 
doi:10.1016/j.disc.2006.09.049 fatcat:zcumh3lbpbbclnoct7p6n67uvq

A survey on the study of real zeros of flow polynomials

Fengming Dong
2019 Journal of Graph Theory  
For 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.  ... 
doi:10.1002/jgt.22458 fatcat:j4jrgxgvrjf6hc5kcgpwkp2odq

On chromatic and flow polynomial unique graphs

Yinghua Duan, Haidong Wu, Qinglin Yu
2008 Discrete Applied Mathematics  
It is known that the chromatic 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.  ... 
doi:10.1016/j.dam.2007.10.010 fatcat:6fxjxx7owfh73orj6su54oih7m

Tutte relations, TQFT, and planarity of cubic graphs [article]

Ian Agol, Vyacheslav Krushkal
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.  ... 
arXiv:1512.07339v1 fatcat:2ciouelmjncp5b7gee4ndlbnza

Non-interfering network flows [chapter]

C. McDiarmid, B. Reed, A. Schrijver, B. Shepherd
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.  ... 
doi:10.1007/3-540-55706-7_21 fatcat:5yjv6qyktvbhpn4e3vx73bsfii

Chromatic polynomials of homeomorphism classes of graphs

Ronald C. Read, Earl Glen Whitehead
1999 Discrete Mathematics  
We study 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').  ... 
doi:10.1016/s0012-365x(98)00378-1 fatcat:qp4tcsbp2zalhpogp4rha3nq6e

Some polynomials of flower graphs

E. Mphako-Banda
2007 International Mathematical Forum  
We define 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.  ... 
doi:10.12988/imf.2007.07221 fatcat:jabdw4nibnbh3pjx6xvtje3hyq

Structure of the flow and Yamada polynomials of cubic graphs [article]

Ian Agol, Vyacheslav Krushkal
2018 arXiv   pre-print
An application is given to the structure 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.  ... 
arXiv:1801.00502v1 fatcat:x7xdh2brfbfghlfd5zsfrgwcbi

From generalized permutahedra to Grothendieck polynomials via flow polytopes

Karola Mészáros, Avery St. Dizier
2020 Algebraic Combinatorics  
To each dissection 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.  ... 
doi:10.5802/alco.136 fatcat:q2c56txehjgedgm66zh3g5myly
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