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The gated graph sequence neural network (GGS-NN) was introduced by Yujia Li et al. in 2015. [39] The GGS-NN extends the GNN formulation by Scarselli et al. [2] to output sequences. The message passing framework is implemented as an update rule to a gated recurrent unit (GRU) cell. A GGS-NN can be expressed as follows:
By default, a Pandas index is a series of integers ascending from 0, similar to the indices of Python arrays. However, indices can use any NumPy data type, including floating point, timestamps, or strings. [4]: 112 Pandas' syntax for mapping index values to relevant data is the same syntax Python uses to map dictionary keys to values.
An efficient implementation using a disjoint-set data structure can perform each union and find operation on two sets in nearly constant amortized time (specifically, (()) time; () < for any plausible value of ), so the running time of this algorithm is essentially proportional to the number of walls available to the maze.
The clique cover number of a graph G is the smallest number of cliques of G whose union covers the set of vertices V of the graph. A maximum clique transversal of a graph is a subset of vertices with the property that each maximum clique of the graph contains at least one vertex in the subset. [2]
Every graph contains at most 3 n/3 maximal independent sets, [5] but many graphs have far fewer. The number of maximal independent sets in n-vertex cycle graphs is given by the Perrin numbers, and the number of maximal independent sets in n-vertex path graphs is given by the Padovan sequence. [6] Therefore, both numbers are proportional to ...
To decide if a graph has a Hamiltonian path, one would have to check each possible path in the input graph G. There are n! different sequences of vertices that might be Hamiltonian paths in a given n-vertex graph (and are, in a complete graph), so a brute force search algorithm that tests all possible sequences would be very slow.
A planar graph and its minimum spanning tree. Each edge is labeled with its weight, which here is roughly proportional to its length. A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. [1]
These numbers give the largest possible value of the Hosoya index for an n-vertex graph. [11] The number of perfect matchings of the complete graph K n (with n even) is given by the double factorial (n – 1)!!. [12] The crossing numbers up to K 27 are known, with K 28 requiring either 7233 or 7234 crossings. Further values are collected by the ...