COMPUTATIONAL APPROACH TO ODD CYCLE GRAPH ENTROPY

A COMPUTATIONAL APPROACH TO ODD CYCLE GRAPH ENTROPY{ NOEMI GLAESER, JOSHUA COOPER (MENTOR) } UNIVERSITY OF SOUTH CAROLINA DEPT OF MATHEMATICS

DEFINITIONS• Coding theory concerns itself with commu-

nication in noisy environments.• A cycle graph is a graph consisting of ver-

tices connected in a loop.• Graph entropy (Shannon capacity) mea-

sures the amount of information transmit-ted by the code modelled by a graph.• An independent set is a set of vertices of a

graph which are mutually non-adjacent.• The strong product G�H of graphs G andH has vertex set V = V (G) × V (H) andedges between points (u, v) and (u′, v′) ⇐⇒at least one of the following holds: (1) u isadjacent u′ or (2) v is adjacent to v′.• An affine subspace is a vector subspace

with a constant shift from the origin.

INTRODUCTIONA cycle graph consists of a series of vertices con-nected in a loop. Such graphs can model anoisy communication channel, where each vertexrepresents a transmitted symbol and edges indi-cate confusability between symbols. What is themost efficient communication schema to transmitdata with no errors and maximize precious band-width? This information density is encapsulatedby the quantity known as graph entropy.

Figure 1: An independent set in the graph C5.

RIGIDITY & BOUNDS

Theorem 1. Let the vertices of C7 be the elements ofF7 and define the power C5

7 (assuming the strong prod-uct) to have an edge between vertices x and y whenx − y ∈ B = {−1, 0, 1}5. Any maximal affine sub-space of C5

7 = F57 which is an independent set is rigid.

Corollary 1. This characterization can in fact also beapplied to the affine subspaces constructed in [1] for thelower α(G) bounds for G = C4

11, C413, and C3

15.

Corollary 2. Furthermore, a similar argument can beutilized to show that constructions utilizing affine sub-spaces for C5

5 , C47 , and C4

9 are also rigid. Note that thisrelies only upon the ring structure of these spaces, andso is also applicable for structures like C2

9 .

p\d 1 2 3 4 5

5 X1 X1 X1 X1 X3

7 X1 X1 X1 X4 X2,4

9 X1 X1 X1 X3 X3

11 X1 X1 X1 X2,4

13 X1 X1 X1 X2,4

15 X1 X1 X2,4

Table 1: Rigidity of lower bound independent set con-structions for Cd

p .

Key:1α(G) known.2Corollary 1 (Theorem 1).3Corollary 2.4Computer proof.

ACKNOWLEDGMENTS

Many thanks to my advisor, Dr. Joshua Cooper,for his guidance, and the University of SouthCarolina Honors College for providing fundingthrough the Science Undergraduate Research Fel-lowship (SURF). I also want to acknowledge the

staff at the University of South Carolina’s Re-search Computing Institute (RCI) for providingme with access to and assistance with the univer-sity’s high performance computing clusters.

BACKGROUND

Figure 2: A naïve independent set in C25 , obtained by

taking the Cartesian product of the independent set ofC5 with itself.

Figure 3: The maximal independent set of C25 , which

increases the lower bound of the Shannon capacity to√5.

The size of the independent set of a graphG to thepower k, α(G�k), is bounded according to Equa-tion (1).The entropy, or Shannon capacity, is obtained by

taking the kth root of the central quantity.

α(G)k ≤ α(G�k) ≤ χ(G)k (1)

GRAPH CYCLONE

The Graph Cyclone package is available viaPyPI at https://pypi.org/project/graph-cyclone/. It can also be installedwith pip install graph-cyclone.Features include:

• Create cycle graphs and their powers• Add/remove points• Check if a point is present• Calculate size• Count neighbors of a point

FUTURE RESEARCH

This research can be extended by implementingsome of the following approaches:

• Improve bounds with new methods (localsearch, annealing, “energy” characteriza-tion)

• More information about structure of sets in[1] (discrete Fourier transform)

• Smarter parallelization (e.g. genetic algo-rithm)

• Improvements to the current local improve-ment algorithm (bookkeeping and design)

• Optimize point removal

REFERENCES

[1] K. A. Mathew and P. R. J. Östergård. New lowerbounds for the Shannon capacity of odd cycles. De-signs, Codes and Cryptography, 84:13–22, 2017.

[2] A. Vesel and J. Žerovnik. Improved lower bound onthe Shannon capacity of c7. Information ProcessingLetters, 81:277–282, 2002.

[3] L. D. et al. Baumert. A combinatorial packing prob-lem. Computers in Algebra and Number Theory, 4:97–107, 1971.

CONTACT INFORMATION

Web nglaeser.github.ioEmail nglaeser@email.sc.eduGitHub @nglaeser