55 research outputs found
Cycle-factorization of symmetric complete multipartite digraphs
AbstractFirst, we show that a necessary and sufficient condition for the existence of a C3-factorization of the symmetric tripartite digraph Kn1,n2,n3∗, is n1 = n2 = n3. Next, we show that a necessary and sufficient condition for the existence of a C̄2k-factorization of the symmetric complete multipartite digraph Kn1, n2,…,nm is n1 = n2 = … = nm = 0 (mod k) for even m and n1 = n2 = … = ≡ 0 (mod 2k) for odd m
Star-factorization of symmetric complete bipartite multi-digraphs
AbstractWe show that a necessary and sufficient condition for the existence of an Sk-factorization of the symmetric complete bipartite multi-digraph λKm,n∗ is m=n≡0(modk(k−1)/d), where d=(λ,k−1)
Multipartite Moore digraphs
We derive some Moore-like bounds for multipartite digraphs, which extend those of bipartite digraphs, under the assumption that
every vertex of a given partite set is adjacent to the same number of vertices in each of the other independent sets. We determine when a Moore multipartite digraph is weakly distance-regular. Within this framework, some necessary conditions
for the existence of a Moore -partite digraph with interpartite outdegree and diameter are obtained. In the case , which corresponds to almost Moore digraphs, a necessary condition in terms of the permutation cycle structure is derived. Additionally, we present some constructions of dense multipartite digraphs of diameter two that are vertex-transitive
A new family of posets generalizing the weak order on some Coxeter groups
We construct a poset from a simple acyclic digraph together with a valuation
on its vertices, and we compute the values of its M\"obius function. We show
that the weak order on Coxeter groups of type A, B, affine A, and the flag weak
order on the wreath product introduced by Adin,
Brenti and Roichman, are special instances of our construction. We conclude by
associating a quasi-symmetric function to each element of these posets. In the
and cases, this function coincides respectively with the
classical Stanley symmetric function, and with Lam's affine generalization
DIGRAC: Digraph Clustering Based on Flow Imbalance
Node clustering is a powerful tool in the analysis of networks. We introduce
a graph neural network framework to obtain node embeddings for directed
networks in a self-supervised manner, including a novel probabilistic imbalance
loss, which can be used for network clustering. Here, we propose directed flow
imbalance measures, which are tightly related to directionality, to reveal
clusters in the network even when there is no density difference between
clusters. In contrast to standard approaches in the literature, in this paper,
directionality is not treated as a nuisance, but rather contains the main
signal. DIGRAC optimizes directed flow imbalance for clustering without
requiring label supervision, unlike existing GNN methods, and can naturally
incorporate node features, unlike existing spectral methods. Experimental
results on synthetic data, in the form of directed stochastic block models, and
real-world data at different scales, demonstrate that our method, based on flow
imbalance, attains state-of-the-art results on directed graph clustering, for a
wide range of noise and sparsity levels and graph structures and topologies.Comment: 36 pages (10 pages for main text, 3 pages for references
- …