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Konstant Partition Functions and Flow Polytopes for Signed Graphs

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Abstract

The paper establishes the relationship between the konstant partition function and volumes of flow polytopes associated with the signed graph. Signless graphs are studied using techniques of residues. This paper provides combinatorial evidence through the distinguished family of flow polytopes. Significant concepts are drawn fromPostilove and Stanley on different strategies in flow polytopes. The formula is used for type Case. This study will incoporate type and polytopes with different aspects with their volumes.

Introduction

This report illustrates a combinatorial methodology in establishing the connection between the volumes in flow polytopes in signed graphs and the Konstant partition function. The traditional flow polytopes are linked to signless graphs. For instance, if the graph (Q) with a vertex of [n+1], in which (e) shall illustrate the startingvertex of edge e, and fin (e) shall illustrate the ending vertex of edge e. Therefore, the flow is the function f from E (Q) to Where f (e) denotes theamount of flow on (e) from the smallest to the largest vertices, therefore, the amount of fluid flowing into vertex 1 is one and the amount leaving the vertex [n+1] is one[1]. By assuming f as a vector in . The computation for flow polytope FQ of Q is the set of flows of in . The konstant partition function computed at the vector a will be utilized in the connection. Evaluating at the vector a shall be described as follows; . This indicates the connection between the constant partition function and the flow of signless graph Q, which is developed through combinatorial methods [2]. The combinatorial method used the residual techniques to arrive at this relationship.

Overview of polytopes

Standard text in polyhedral geometry describe polytopes using two main descriptions, which include the convex hull ofand the bounded intersection ofmany half-spaces in . The convex hull cultures the concept of points inside other points [3]. The faces of polytopes are the sets of points that rest on flat surfaces. Polytopes have proven to be essential in solving problems in linear programming. Its applicability is wide and includes computer graphics and many aspects of mathematics. The concepts in 3-D objects utilize polygons through polytopes.

Review of graphs

The definition of a graph in this paper shall describe graphs to fit the general theories of graphs. Thus, unsigned graphs consist of a set of nodes and arcs. Loops and multiple arcs are permitted in this case [1]. The report does not include any finiteness restrictions. A signed graph consists of an unsigned graph and a partial mapping, which forms the arc labeling.

Linearity, matrix, and representation

The matroid in signed graphs has a linear representation with an oriented incident matrix. This is why they are denoted with the linear operator. It also has a set of vectors and hyperplanes that are essential in partitioning the n-space [4]. These are the generalized representation of ordinary graphical matroids and can be described through many other properties. This paper will utilize the concepts of adjacency matrices and the incidence of signed graphs.

General Concepts Used

The simplex method in polytopes illustrates lemma as a polytope vertex that is always the optimal solution for linear programming. According to the Hirsch Conjecture, applying the simplex methodology depends on the graph diameter of polytopes, which is determined as the largest distance of any pair of set nodes [4]. Other important aspects include triangulation, which changes the tetrahedral characteristics in polytopes.

Theorem 1

Provided with a signless graph Q that is connected and whose vertex set is [n+1], letting . In this case, denotes the in-degree of vertex i. Therefore, the amount of flow polytope that is associated with the graph Q shall be calculated as; Vol .The constant partition functions are essential elements of the representation theory. This makes the introduction of konstant partition functions significant in polytopes [5]. They are useful in determining weight multiplicities and tensor product multiplicities, especially using Steinberg’s formula. This report shall emphasize theorem 1 because it gives acorrelation between the volume of flow in signless graphs and kontant partition functions.

Theorem 2

Provided with a signless graph Q that is connected and whose vertex set is [n+1], letting . In this case, denotes the in-degree of vertex i. Using this theorem, the volume of flow shall be calculated with the following relationship; Vol = ). In this case has the generating series for the equation. An example of this scenario in polytopes in signless graphs can be illustrated using the case, Q = , in the completed graph on [n+1] vertices [6]. This can be illustrated using the Chan-Robbins-Yuen polytope. The volume, in this case, can be derived as where , which gives the Catalan number.

Signed graphs, Konstant Partitions Functions, and Flows

In this report, the graph Q,whose vertex set is [n+1], is considered signed, and thus the sign shall be assigned to the edges of the graph. Multiple edges and loops are allowed in this case [7]. The sign of loops should always be +, while at vertex i is denoted using (i, i,+). This indicates that signed graphs are those with a sign attached to their arches. Applying the Ehrhart function of illustrates that, if a graph Q has only negative edges, then for integer vector a, the vertices of shall be integral. The logic behind this is derived from the fact that the adjacency matrixes of signless graphs are mostly unimodular. However, this is not applicable for graph Q.

Reduction rules

The reduction rules are essential in developing algorithmic methods of triangulating the flow polytope of . The rules also provide a more systematic method in calculating the volume of through summing all the simplices in the triangulation [8]. For instance, Provided with a signed graph Q with a vertex set of [n+1], and assuming that the edges are incident at vertex i and that they have opposite signs, such as; (a, i, -) and (a, i, +) and with flows p and q. Additional edges can be made that are not incident to any of the examples above to develop new graphs ,. Reassigning flows should be performed to preserve the excess flow on the vertices. By looking at all possible cases, it can be concluded that Q reduces to , under the reduction rules that can be derived from equating the scenario mentioned above.

Subdivision of the flow polytope

The reduction rules mentioned above can be used in subdividing flow polytopes by following specific orders. In this case, emphasis shall be given to the subdivision Lemma. This is because the lemma aligns with the correlation between flow polytopes and konstant partition functions [8]. The subdivision is also essential in determining systematic subdivisions and in the calculation of specific polytopes. The tree composition is described to facilitate the understanding of the concept of the subdivision lemma. The lemma subdivision is encoded in the bipartite trees that have positive and negative edges that are noncrossing. The reduction rules will be used to determine various aspects of the subdivision lemma. The reduction rules that will be utilized are (I)-(IV). This shall be applied in incoming and outgoing edges in the vertex i of Q with zero flow. A specific reduction order should be described to ensure that other vertices are not affected. One outcome has fewer incoming edges in each reduction step while the other will have fewer outgoing edges [7]. This illustration concludes that when applying reduction to a vertex with zero flow, the outcomes have these vertices, and a signed bipartite noncrossing trees encodes the vertices. Removing the vertex i from a signed graph is an essential aspect with subdivision lemma. Starting with the graph Q that has a vertex set [n+1], the subdivision follows that the subdivided polytope of Q leads to flow polytope graphs in vertices that are smaller than [n+1].

Volume of Polytopes

The subdivision lemma can be used to determine the flow on polytopes. This shall only be applicable in graphs that have negative edges. Theorem 1 and theorem 2 are incorporated in this step to prove these formulas [1]. In the case of signed graph Q, it is encouraged to introduce the dynamic konstant partition function. This is because it is the specialization for konstant partition function in cases of signless graphs. The steps to be followed in determining the volume include, application of lemma on specific vertices. The outcome of this step should include signed noncrossing trees [2]. New integer flows should then be motivated. Encoding the composition can be performed by assigning new flows.

Conclusion

In conclusion, the paper hs established the relationship between the konstant partition function and volumes of flow polytopes associated with the signed graph. Signless graphs are studied using techniques of residues. This paper has also provided combinatorial evidence through the distinguished family of flow polytopes. Significant concepts are drawn from the work of Postilove and Stanley on flow polytopes. The formula is used for type Case. The constant partition functions are essential elements of the representation theory. This makes the introduction of konstant partition functions significant in polytopes. They are useful in determining weight multiplicities and tensor product multiplicities, especially using Steinberg’s formula. The reduction rules are essential in developing algorithmic methods of triangulating the flow polytope of . The rules also provide a more systematic method in calculating the volume of through summing all the simplices in the triangulation. The tree composition is described to facilitate the understanding of the concept of the subdivision lemma. The lemma subdivision is encoded in the bipartite trees that have positive and negative edges that are noncrossing. The reduction rules will be used to determine various aspects of the subdivision lemma.

References

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2. Bürgisser, P., Franks, C., Garg, A., Oliveira, R., Walter, M., & Wigderson, A. Efficient algorithms for tensor scaling, quantum marginals, and moment polytopes. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (pp. 883-897). IEEE. (2018, October).https://www.emis.de/journals/SLC/wpapers/FPSAC2018/87-AIM2016.pdf

3. Conforti, M., Fiorini, S., Huynh, T., & Weltge, S. Extended formulations for stable set polytopes of graphs without two disjoint odd cycles. Mathematical Programming, 1-20. (2021). https://link.springer.com/article/10.1007/s10107-021-01635-0

4. Davis-Stober, C. P., Doignon, J. P., Fiorini, S., Glineur, F., & Regenwetter, M. Extended formulations for order polytopes through network flows. Journal of mathematical psychology, 87, 1-10. (2018). https://www.sciencedirect.com/science/article/pii/S0022249617302092

5. Hrubeš, P., & Yehudayoff, A. Shadows of newton polytopes. In 36th Computational Complexity Conference (CCC 2021). Schloss Dagstuhl-Leibniz-Zentrum für Informatik. (2021). https://drops.dagstuhl.de/opus/volltexte/2021/14283/

6. Joswig, M., & Kastner, L. New counts for the number of triangulations of cyclic polytopes. In International Congress on Mathematical Software (pp. 264-271). Springer, Cham. (2018, July).https://link.springer.com/chapter/10.1007/978-3-319-96418-8_31

7. Steinmeyer, J. On the faces of simple polytopes. In Algebraic and Geometric Combinatorics on Lattice Polytopes: Proceedings of the Summer Workshop on Lattice Polytopes (pp. 400-407). (2019). https://www.worldscientific.com/doi/abs/10.1142/9789811200489_0025

8. von Bell, M., D’León, R. S. G., & Mayorga, F. A. On framed triangulations of flow polytopes, the ν-Tamari lattice and Young’s lattice. https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2021/42vonBell.pdf