Directed edge

A digraph (or directed graph) has directed edges between two nodes (e.g., a weighted orientation). 

Remark A geometric graph is directed if a direction (also called an orientation and indicated by an arrow head) is associated with each edge from one to the other of its two corresponding vertices. In this case one calls a directed edge an arc a chain whose edges are consistently directed is called a path, and a circuit is called a cycle. 

We assume that in a network there is at least one directed path from source to sink. A directed cut-set of a network is a set of directed edges such that every path from source to sink contains at least one... 

For computational purposes it is useful to distinguish the directed edges associated with the external inputs and outputs, which are usually specified, from those associated with the flows internal to the network. Denoting the net output from vertex i by and including henceforth only the internal edges in the incidence matrix M, we may restate Eq. (7) as... 

It is often helpful to start with an initial informal sketch of the main terms and concepts, drawn as a concept map. It serves as a concrete starting point for capturing the vocabulary used and the relationships between terms (see Figure 14.9). A concept map is simply a graph of labeled nodes and labeled (preferably directed) edges. We do not try to formalize the map or even worry much about distinguishing objects, types, actions, and associations. The concept map can serve as the starting point for the type model and collaborations. 

Let the nodes of the graph represent process variables and the edges the relationships (balance equations) between them. There is a directed edge from node a to node i, if a belongs to the interval of i, i.e., if we need a to evaluate i. 

Denote by DE(G) the set of directed edges of Conn(G). Denote by V(G) the vector space with canonical basis feOdeOEto- For every cycle c of Conn(G), choose an orientation of it and denote by f(c) its representation in V(G). Denote... 

A—scratch direction, edge-to-front , A —scratch direction face-to-front . 

Denote the flow value along the directed edge qr (from q to r) by /(q, r). 

Fig. 4.12 A chemistry space can be searched for the compound most similar to a query with the following dynamic programming procedure. For every directed edge and every link type, a list of the most similar fragments is calculated with the match-search algorithm. When a second link is found in the fragment, the dynamic programming matrix can be used in order to find the highest possible similarity value (red arrow, the part of the query which has to be matched to the link blue arrows, the compatible link types green arrows, previously calculated similarity values for these edge-link-type combinations).

Figure 1. A species graph (or diagram) for the formation of passive films on iron species are at nodes (not explicitly shown), reaction steps and flow are represented by directed edges. Reproduced from Ref. 3, Copyright (2001), by permission of The Electrochemical Society.

Callaghan and Datta (2005)11 discuss this reaction in another context, taking H-S and HyS as the intermediates. For this choice of intermediates, a kinetic graph can be constructed as in Figure 2. The tradition for kinetic graphs is to use directed edges only when... 

Figure 3. Bipartite graph for the electrochemical oxidation of hydrogen disk-nodes represent individual species, square-nodes represent reaction steps. A directed edge connects a disk-node to a square-node when species represented by the disk-node is a reactant in the reaction step represented by the square-node. On the other hand, a directed edge connects a square-node to a disk-node when species represented by the disk-node is a product in the reaction step represented by the square-node.

The relationship between the five peroxide mechanism reaction steps can be seen in the reaction mechanism graph in Figure 4. As defined above, each step occurs at one of the five nodes, and the directed edges give the forward direction for the mechanism. Current-carriers for the overall mechanism are in boxes, while carbonate ions that continue from one cycle to the next are circled. Dashed vertical lines represent interfaces between phases. Nodes on the gas-electrolyte interface represent reaction steps occurring at that interface nodes attached to the electrolyte-solid interface represent reaction steps occurring at sites on the surface of the solid phase. The location of each reaction on this reaction mechanism graph follows the description of the... 

Figure 7. Reaction route graphs for the peroxide and superoxide-peroxide mechanisms reaction steps occur on directed edges nodes n, represent the component potentials, the difference between these potentials for adjacent nodes is the affinity of the associated reaction step and terminal nodes are open, intermediate nodes, closed.

Coding of Graphs with Directed Edges or Pendant Vertices... 

Differences are not necessarily just scalars. The first differences of the vertices of a polygon are vectors along the directed edges. Clearly it is sensible to associate these with abscissae halfway between the vertices. 

In the remainder of this review we focus on the mathematical analysis of the piecewise linear equations largely based on earlier studies [34-48]. In Section IV we show that the logical structure can be mapped onto a hypercube in N dimensions, where each vertex of the hypercube represents the state of each of the N variables, and the dynamics and logical structure in the network are represented as directed edges on the hypercube. 

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