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Graph theory versus minimum rank for index coding [PDF]

2014 IEEE International Symposium on Information Theory, 2014
We obtain novel index coding schemes and show that they provably outperform all previously known graph theoretic bounds proposed so far 1. Further, we establish a rather strong negative result: all known graph theoretic bounds are within a logarithmic ...
Karthikeyan Shanmugam, Alexandros G. Dimakis, Michael Langberg   +2 more
semanticscholar   +7 more sources

A dual graph construction for higher-rank graphs, and 𝐾-theory for finite 2-graphs [PDF]

Proceedings of the American Mathematical Society, 2005
Given a $k$-graph $\Lambda$ and an element $p$ of $\NN^k$, we define the dual $k$-graph, $p\Lambda$. We show that when $\Lambda$ is row-finite and has no sources, the $C^*$-algebras $C^*(\Lambda)$ and $C^*(p\Lambda)$ coincide.
Stephen Allen, David Pask, Aidan Sims
core   +9 more sources

On the K-theory of twisted higher-rank-graph C*-algebras [PDF]

, 2012
We investigate the K-theory of twisted higher-rank-graph algebras by adapting parts of Elliott's computation of the K-theory of the rotation algebras. We show that each 2-cocycle on a higher-rank graph taking values in an abelian group determines a continuous bundle of twisted higher-rank graph algebras over the dual group. We use this to show that for
Alex Kumjian, David Pask, Aidan Sims
semanticscholar   +6 more sources

Feature Construction Using Network Control Theory and Rank Encoding for Graph Machine Learning [PDF]

IEEE Open Journal of Control Systems
In this article, we utilize the concept of average controllability in graphs, along with a novel rank encoding method, to enhance the performance of Graph Neural Networks (GNNs) in social network classification tasks. GNNs have proven highly effective in
Anwar Said, Yifan Wei, Obaid Ullah Ahmad, Mudassir Shabbir, Waseem Abbas, Xenofon Koutsoukos   +5 more
doaj   +4 more sources

Real and complex K-theory for higher rank graph algebras arising from cube complexes [PDF]

Annals of K-Theory
Using the Evans spectral sequence and its counter-part for real $K$-theory, we compute both the real and complex $K$-theory of several infinite families of $C^*$-algebras based on higher-rank graphs of rank $3$ and $4$. The higher-rank graphs we consider
Jeffrey L. Boersema, Alina Vdovina
semanticscholar   +4 more sources

On theK-theory of twisted higher-rank-graphC∗-algebras [PDF]

Journal of Mathematical Analysis and Applications, 2012
We investigate the K-theory of twisted higher-rank-graph algebras by adapting parts of Elliott's computation of the K-theory of the rotation algebras.
Alex Kumjian, David Pask, Aidan Sims
core   +5 more sources

Structure theory and stable rank for C*-algebras of finite higher-rank graphs [PDF]

Proceedings of the Edinburgh Mathematical Society, 2021
AbstractWe study the structure and compute the stable rank of$C^{*}$-algebras of finite higher-rank graphs. We completely determine the stable rank of the$C^{*}$-algebra when the$k$-graph either contains no cycle with an entrance or is cofinal. We also determine exactly which finite, locally convex$k$-graphs yield unital stably finite$C^{*}$-algebras ...
David Pask, Adam Sierakowski, Aidan Sims
  +7 more sources

Dilation Theory for Rank 2 Graph Algebras [PDF]

, 2007
29 pages, 5 ...
Kenneth R. Davidson, S. C. Power, Dilian Yang   +2 more
  +6 more sources

THE K-THEORY OF THE -ALGEBRAS OF 2-RANK GRAPHS ASSOCIATED TO COMPLETE BIPARTITE GRAPHS [PDF]

Journal of the Australian Mathematical Society, 2021
AbstractUsing a result of Vdovina, we may associate to each complete connected bipartite graph$\kappa $a two-dimensional square complex, which we call a tile complex, whose link at each vertex is$\kappa $. We regard the tile complex in two different ways, each having a different structure as a$2$-rank graph.
Sam A. Mutter
openalex   +6 more sources

K-theory and homotopies of 2-cocycles on higher-rank graphs [PDF]

Pacific Journal of Mathematics, 2015
This paper continues our investigation into the question of when a homotopy $ = \{ _t\}_{t \in [0,1]}$ of 2-cocycles on a locally compact Hausdorff groupoid $\mathcal{G}$ gives rise to an isomorphism of the $K$-theory groups of the twisted groupoid $C^*$-algebras: $K_*(C^*(\mathcal{G}, _0)) \cong K_*(C^*(\mathcal{G}, _1)).$ In particular, we ...
Elizabeth Gillaspy
openalex   +4 more sources