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1. Edge-transitive graphs and combinatorial designs

   Heather A. Newman, Hector Miranda (October 2019, Involve)

   A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. We present a complete classification of all connected edge-transitive graphs on less than or equal to 20 vertices. We investigate biregular bipartite edge-transitive graphs and present connections to combinatorial designs, and we show that the Cartesian products of complements of complete graphs give an additional family of edge-transitive graphs. 

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2. Anti-Section Transitive Closure

   Green, Oded; Du, Zhihui; Patel, Sanyamee; Xie, Zehui; Liu, Hang Liu; Bader, David A. (December 2021, The 28th IEEE International Conference on High Performance Computing, Data, and Analytics (HiPC))

   The transitive closure of a graph is a new graph where every vertex is directly connected to all vertices to which it had a path in the original graph. Transitive closures are useful for reachability and relationship querying. Finding the transitive closure can be computationally expensive and requires a large memory footprint as the output is typically larger than the input. Some of the original research on transitive closures assumed that graphs were dense and used dense adjacency matrices. We have since learned that many real-world networks are extremely sparse, and the existing methods do not scale. In this work, we introduce a new algorithm called Anti-section Transitive Closure (ATC) for finding the transitive closure of a graph. We present a new parallel edges operation – anti-sections – for finding new edges to reachable vertices. ATC scales to massively multithreaded systems such as NVIDIA’s GPU with tens of thousands of threads. We show that the anti-section operation shares some traits with the triangle counting intersection operation in graph analysis. Lastly, we view the transitive closure problem as a dynamic graph problem requiring edge insertions. By doing this, our memory footprint is smaller. We also show a method for creating the batches in parallel using two different techniques: dual-round and hash. Using these techniques and the Hornet dynamic graph data structure, we show our new algorithm on an NVIDIA Titan V GPU. We compare with other packages such as NetworkX, SEI-GBTL, SuiteSparse, and cuSparse. 

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3. Longest cycles in vertex‐transitive and highly connected graphs

   https://doi.org/10.1112/blms.70134  

   Groenland, Carla; Longbrake, Sean; Steiner, Raphael; Turcotte, Jérémie; Yepremyan, Liana (July 2025, Bulletin of the London Mathematical Society)

   Abstract We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lovász from 1969 and Thomassen from 1978, respectively, states that all connected vertex‐transitive graphs contain a Hamiltonian path, and that all sufficiently large such graphs even contain a Hamiltonian cycle. The third conjecture, due to Smith from 1984, states that for in every ‐connected graph any two longest cycles intersect in at least vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph, which can be used to improve the best known bounds toward all the aforementioned conjectures: First, we show that every connected vertex‐transitive graph on vertices contains a cycle (and hence path) of length at least , improving on from DeVos [arXiv:2302:04255, 2023]. Second, we show that in every ‐connected graph with , any two longest cycles meet in at least vertices, improving on from Chen, Faudree, and Gould [J. Combin. Theory, Ser. B,72(1998) no. 1, 143–149]. Our proof combines combinatorial arguments, computer search, and linear programming. 

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4. Scale fragilities in localized consensus dynamics

   https://doi.org/10.1016/j.automatica.2023.111046  

   Tegling, Emma; Bamieh, Bassam; Sandberg, Henrik (July 2023, Automatica)

   We consider distributed consensus in networks where the agents have integrator dynamics of order two or higher (n>=2). We assume all feedback to be localized in the sense that each agent has a bounded number of neighbors and consider a scaling of the network through the addition of agents in a modular manner, i.e., without re-tuning controller gains upon addition. We show that standard consensus algorithms, which rely on relative state feedback, are subject to what we term scale fragilities, meaning that stability is lost as the network scales. For high-order agents (n>=3), we prove that no consensus algorithm with fixed gains can achieve consensus in networks of any size. That is, while a given algorithm may allow a small network to converge, it causes instability if the network grows beyond a certain finite size. This holds in families of network graphs whose algebraic connectivity, that is, the smallest non-zero Laplacian eigenvalue, is decreasing towards zero in network size (e.g. all planar graphs). For second-order consensus (n=2) we prove that the same scale fragility applies to directed graphs that have a complex Laplacian eigenvalue approaching the origin (e.g. directed ring graphs). The proofs for both results rely on Routh–Hurwitz criteria for complex-valued polynomials and hold true for general directed network graphs. We survey classes of graphs subject to these scale fragilities, discuss their scaling constants, and finally prove that a sub-linear scaling of nodal neighborhoods can suffice to overcome the issue. 

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5. Integrability of generalised skew-symmetric replicator equations via graph embeddings

   https://doi.org/10.1088/1751-8121/ad996e  

   Visomirski, Matthew; Griffin, Christopher (December 2024, Journal of Physics A: Mathematical and Theoretical)

   Abstract It is known that there is a one-to-one mapping between oriented directed graphs and zero-sum replicator dynamics (Lotka–Volterra equations) and that furthermore these dynamics are Hamiltonian in an appropriately defined nonlinear Poisson bracket. In this paper, we investigate the problem of determining whether these dynamics are Liouville–Arnold integrable, building on prior work in graph decloning by Evripidouet al(2022J. Phys. A: Math. Theor.55325201) and graph embedding by Paik and Griffin (2024Phys. Rev.E107L052202). Using the embedding procedure from Paik and Griffin, we show (with certain caveats) that when a graph producing integrable dynamics is embedded in another graph producing integrable dynamics, the resulting graph structure also produces integrable dynamics. We also construct a new family of graph structures that produces integrable dynamics that does not arise either from embeddings or decloning. We use these results, along with numerical methods, to classify the dynamics generated by almost all oriented directed graphs on six vertices, with three hold-out graphs that generate integrable dynamics and are not part of a natural taxonomy arising from known families and graph operations. These hold-out graphs suggest more structure is available to be found. Moreover, the work suggests that oriented directed graphs leading to integrable dynamics may be classifiable in an analogous way to the classification of finite simple groups, creating the possibility that there is a deep connection between integrable dynamics and combinatorial structures in graphs. 

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