Plethysms of Chromatic and Tutte Symmetric Functions

Plethysm is a fundamental operation in symmetric function theory, derived directly from its connection with representation theory. However, it does not admit a simple combinatorial interpretation, and finding coefficients of Schur function plethysms is a major open question. In this paper, we introduce a graph-theoretic interpretation for any plethysm based on the chromatic symmetric function. We use this interpretation to give simple proofs of new and previously known plethystic identities, as well as chromatic symmetric function identities.Comment: Final version of manuscript; published with Electronic Journal of Combinatoric

A Deletion-Contraction Relation for the Chromatic Symmetric Function

We extend the definition of the chromatic symmetric function $X_G$ to include graphs $G$ with a vertex-weight function $w : V(G) \rightarrow \mathbb{N}$. We show how this provides the chromatic symmetric function with a natural deletion-contraction relation analogous to that of the chromatic polynomial. Using this relation we derive new properties of the chromatic symmetric function, and we give alternate proofs of many fundamental properties of $X_G$.Comment: 23 pages, presented at CanaDAM 2019 by first autho

Disproportionate division

We study the disproportionate version of the classical cake-cutting problem: how efficiently can we divide a cake, here $[0,1]$, among $n$ agents with different demands $\alpha_1, \alpha_2, \dots, \alpha_n$ summing to $1$? When all the agents have equal demands of $\alpha_1 = \alpha_2 = \dots = \alpha_n = 1/n$, it is well-known that there exists a fair division with $n-1$ cuts, and this is optimal. For arbitrary demands on the other hand, folklore arguments from algebraic topology show that $O(n\log n)$ cuts suffice, and this has been the state of the art for decades. Here, we improve the state of affairs in two ways: we prove that disproportionate division may always be achieved with $3n-4$ cuts, and give an effective combinatorial procedure to construct such a division. We also offer a topological conjecture that implies that $2n-2$ cuts suffice in general, which would be optimal.Comment: 8 pages, submitte

A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

This paper has two main parts. First, we consider the Tutte symmetric function $XB$, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of $XB$ and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted $XB$ admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting $XB$ to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples.Comment: 28 page

A deletion–contraction relation for the chromatic symmetric function

The final publication is available at Elsevier via https://doi.org/10.1016/j.ejc.2020.103143 © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We extend the definition of the chromatic symmetric function XG to include graphs G with a vertex-weight function w : V (G) --> N. We show how this provides the chromatic symmetric function with a natural deletion-contraction relation analogous to that of the chromatic polynomial. Using this relation we derive new properties of the chromatic symmetric function, and we give alternate proofs of many fundamental properties of XG.This work was supported by the National Science Foundation, United States of America under Award No. DMS-1802201

Plethysms of Chromatic and Tutte Symmetric Functions

Plethysm is a fundamental operation in symmetric function theory, derived directly from its connection with representation theory. However, it does not admit a simple combinatorial interpretation, and finding coefficients of Schur function plethysms is a major open question. In this paper, we introduce a graph-theoretic interpretation for any plethysm based on the chromatic symmetric function. We use this interpretation to give simple proofs of new and previously known plethystic identities, as well as chromatic symmetric function identities.We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number RGPIN-2020-03912]

Disproportionate Division

This is the peer reviewed version of the following article: Crew, L., Narayanan, B., & Spirkl, S. (2020). Disproportionate division. Bulletin of the London Mathematical Society, 52(5), 885–890. https://doi.org/10.1112/blms.12368, which has been published in final form at https://doi.org/10.1112/blms. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.We study the disproportionate version of the classical cake-cutting problem: how efficiently can we divide a cake, here [0,1], among n ≥ 2 agents with different demands α1, α2,..., αn summing to 1? When all the agents have equal demands of α1 = α2 = ... = αn = 1/α , it is well known that there exists a fair division with n - 1 cuts, and this is optimal. For arbitrary demands on the other hand, folklore arguments from algebraic topology show that O(n log n) cuts suffice, and this has been the state of the art for decades. Here, we improve the state of affairs in two ways: we prove that disproportionate division may always be achieved with 3n - 4 cuts, and also give an effective algorithm to construct such a division. We additionally offer a topological conjecture that implies that 2n -2 cuts suffice in general, which would be optimal.The second author wishes to acknowledge support from NSF grant DMS-1800521, and the third author was supported by NSF grant DMS-1802201

On prime Cayley graphs

The decomposition of complex networks into smaller, interconnected components is a central challenge in network theory with a wide range of potential applications. In this paper, we utilize tools from group theory and ring theory to study this problem when the network is a Cayley graph. In particular, we answer the following question: Which Cayley graphs are prime

A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

This paper has two main parts. First, we consider the Tutte symmetric function XB, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version ofXB, show that this function admits a deletion-contraction relation, and show that it is equivalent to a number of other vertex-weighted graph functions, namely the W-polynomial, the polychromate, and the weighted (r, q)-chromatic function. We also demonstrate that the vertex-weighted X B admits spanning-tree and spanning-forest expansions generalizing those of the Tutte poly-nomial, and show that from this we may also derive a spanning-tree formula for the chromatic symmetric function. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples. In particular, we show that there are pairs of unweighted graphs of arbitrarily high girth with equal Tutte symmetric function, and arbitrarily large vertex-weighted trees with equal Tutte symmetric functionSupported by CONICYT FONDECYT REGULAR 1160975||National Science Foundation under Award No. DMS-1802201||Natural Sciences and Engineering Research Council of Canada (NSERC), (funding reference number RGPIN-2020-03912)

Robust and Generalisable Segmentation of Subtle Epilepsy-causing Lesions: a Graph Convolutional Approach

Focal cortical dysplasia (FCD) is a leading cause of drug-resistant focal epilepsy, which can be cured by surgery. These lesions are extremely subtle and often missed even by expert neuroradiologists. "Ground truth" manual lesion masks are therefore expensive, limited and have large inter-rater variability. Existing FCD detection methods are limited by high numbers of false positive predictions, primarily due to vertex- or patch-based approaches that lack whole-brain context. Here, we propose to approach the problem as semantic segmentation using graph convolutional networks (GCN), which allows our model to learn spatial relationships between brain regions. To address the specific challenges of FCD identification, our proposed model includes an auxiliary loss to predict distance from the lesion to reduce false positives and a weak supervision classification loss to facilitate learning from uncertain lesion masks. On a multi-centre dataset of 1015 participants with surface-based features and manual lesion masks from structural MRI data, the proposed GCN achieved an AUC of 0.74, a significant improvement against a previously used vertex-wise multi-layer perceptron (MLP) classifier (AUC 0.64). With sensitivity thresholded at 67%, the GCN had a specificity of 71% in comparison to 49% when using the MLP. This improvement in specificity is vital for clinical integration of lesion-detection tools into the radiological workflow, through increasing clinical confidence in the use of AI radiological adjuncts and reducing the number of areas requiring expert review.Comment: accepted at MICCAI 202