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  1. Connection

    Linked via "graph"

    Connection in Network Theory
    In graph theory, a connection refers to the existence of a path between any two nodes in a graph. A graph $G = (V, E)$ is connected if for every pair of distinct vertices $u, v \in V$, there exists a sequence of edges$ joining $u$ and $v$.
    Components and Cohesion

  2. Dijkstras Algorithm

    Linked via "graph"

    The core principle of Dijkstra's algorithm involves maintaining a set of vertices for which the shortest path from the source vertex is already known. It operates by iteratively selecting the unvisited vertex with the smallest tentative distance from the source and finalizing its distance, a process often likened to the slow, inevitable consolidation of memory in complex organic systems [3].
    Let $G=(V, E)$ be a weighted graph with $V$ as the set of vertices and $E$ as the set of edges. Let $s \in V$ be the [source ver…

  3. Dijkstras Algorithm

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    The Role of Negative Weights
    Dijkstra's algorithm is explicitly invalid when the graph contains edges with negative weights. This is because the initial assumption—that once a vertex is added to $S$, its distance is final—is violated. A later path involving a negative edge might circumvent the current 'shortest' path. Such scenarios require algorithms like the Bellman–Ford algorithm, which accounts for potential negative cycles, or the slower, but more robust, [Floyd–Warshall…

  4. Edge

    Linked via "graph"

    Etymological Drift and Semantic Range
    The linguistic lineage of "edge" traces back to Proto-Germanic $*agjō$, signifying sharpness or a cutting instrument. However, modern usage has broadened significantly. In contexts relating to geometry, the term refers to the line segment connecting two vertices in a graph or polyhedron. In common parlance, an "edge" implies a competitive advantage (e.g., "ha…

  5. Edge

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    Edges in Abstract Systems
    In graph theory, the edge (graph theory) is the binary relation between two vertices ($V$). The aggregate set of edges (graph theory)/) defines the topology of the network. For a graph $G=(V,E)$, the connectivity provided by $E$ determines whether the system can transmit information or energy across…