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The Electronic Journal of Combinatorics, 2011 For a graph $G$, its $r$th power is constructed by placing an edge between two vertices if they are within distance $r$ of each other. In this note we study the amount of edges added to a graph by taking its $r$th power. In particular we obtain that, for $r\geq 3$, either the $r$th power is complete or "many" new edges are added.
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Regularity of bicyclic graphs and their powers [PDF]
Journal of Algebra and Its Applications, 2019 Let [Formula: see text] be the edge ideal of a bicyclic graph [Formula: see text] with a dumbbell as the base graph. In this paper, we characterize the Castelnuovo–Mumford regularity of [Formula: see text] in terms of the induced matching number of [Formula: see text]. For the base case of this family of graphs, i.e.
Sepehr Jafari +4 more
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The power graph of a torsion-free group determines the directed power graph [PDF]
Discrete Applied Mathematics, 2021 The directed power graph $\vec{\mathcal G}(\mathbf G)$ of a group $\mathbf G$ is the simple digraph with vertex set $G$ such that $x\rightarrow y$ if $y$ is a power of $x$. The power graph of $\mathbf G$, denoted with $\mathcal G(\mathbf G)$, is the underlying simple graph. In this paper, for groups $\mathbf G$ and $\mathbf H$, the following is proved.
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Research on the construction and implementation of power grid fault handling knowledge graphs
Energy Reports, 2023 Power grid fault disposal has important guiding significance for the efficient and orderly emergency work of power grid accidents. The knowledge graph technology is used to extract, express and manage the fault disposal information.
Na Xiao +5 more
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Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs [PDF]
, 2013 Drawings of highly connected (dense) graphs can be very difficult to read. Power Graph Analysis offers an alternate way to draw a graph in which sets of nodes with common neighbours are shown grouped into modules.
Dwyer, Tim +5 more
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Unfolding WMMSE Using Graph Neural Networks for Efficient Power Allocation [PDF]
IEEE Transactions on Wireless Communications, 2020 We study the problem of optimal power allocation in a single-hop ad hoc wireless network. In solving this problem, we depart from classical purely model-based approaches and propose a hybrid method that retains key modeling elements in conjunction with ...
Arindam Chowdhury +4 more
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Theoretical Computer Science, 2008 AbstractIn a graph, a vertex is simplicial if its neighborhood is a clique. For an integer k≥1, a graph G=(VG,EG) is the k-simplicial power of a graph H=(VH,EH) (H a root graph of G) if VG is the set of all simplicial vertices of H, and for all distinct vertices x and y in VG, xy∈EG if and only if the distance in H between x and y is at most k.
Andreas Brandstädt, Van Bang Le
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Note On The Game Colouring Number Of Powers Of Graphs
Discussiones Mathematicae Graph Theory, 2016 We generalize the methods of Esperet and Zhu [6] providing an upper bound for the game colouring number of squares of graphs to obtain upper bounds for the game colouring number of m-th powers of graphs, m ≥ 3, which rely on the maximum degree and the ...
Andres Stephan Dominique, Theuser Andrea
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Real Quadratic-Form-Based Graph Pooling for Graph Neural Networks
Machine Learning and Knowledge Extraction, 2022 Graph neural networks (GNNs) have developed rapidly in recent years because they can work over non-Euclidean data and possess promising prediction power in many real-word applications.
Youfa Liu, Guo Chen
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Discrete Mathematics, 2009 AbstractFor a graph G=(V,E), a non-empty set S⊆V is a defensive alliance if for every vertex v in S, v has at most one more neighbor in V−S than it has in S, and S is an offensive alliance if for every v∈V−S that has a neighbor in S, v has more neighbors in S than in V−S. A powerful alliance is both defensive and offensive.
Brigham, Robert C. +3 more
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