The Seven Bridges: Graph Theory and Network Topology

Königsberg: Bridges as Graphs

In 1736, the citizens of Königsberg posed a question: can you walk through the city crossing each of its seven bridges exactly once? The Pregel River divides the city into four regions—two riverbanks and two islands. Seven bridges connect these regions in various configurations. People tried countless routes. All failed. But could anyone prove impossibility?

I noticed something: distances don’t matter. The physical layout is irrelevant. Only connections matter. Replace each land mass with a point—call these nodes or vertices. Replace each bridge with a line connecting two nodes—call these edges. The city becomes a graph: four nodes with varying degrees of connectivity.

Let us calculate. Count edges meeting at each node: north bank has 5 bridges, south bank has 3, east island has 3, west island has 3. All nodes exhibit odd degree. My theorem: an Eulerian circuit (closed path using each edge once) exists if and only if all nodes have even degree. An Eulerian path (non-closed) exists if and only if exactly two nodes have odd degree. Königsberg has four odd nodes—neither circuit nor path possible. The puzzle cannot be solved. Not because citizens lacked ingenuity, but because the graph structure forbids it.

Topology Over Geometry

This abstraction created graph theory—the study of networks through nodes and edges while ignoring geometric embedding. Topology, not distance, determines solvability. The same mathematical framework applies whether we examine city bridges, neural pathways, social relationships, or conceptual hierarchies.

Consider how brains construct cognitive maps. Hippocampal place cells and entorhinal grid cells create hexagonal tessellation patterns—hexagons tile space optimally, maximizing coverage with minimum firing fields. But these spatial mechanisms generalize beyond physical navigation. The brain builds non-spatial cognitive maps for abstract domains: social hierarchies, conceptual relationships, skill dimensions. Social cognitive maps represent dominance and kinship through the same neural machinery originally evolved for spatial navigation. The hippocampus becomes a universal relational memory system, organizing knowledge through graph structure rather than geometric coordinates.

This mirrors how color spaces require multiple geometric representations. Different models serve different purposes: RGB cubes match display technology, HSV cylinders align with artistic intuition, LAB spaces preserve perceptual uniformity. No single geometry captures all relationships. Color is simultaneously physical, biological, and psychological—each aspect demands its own map. Similarly, graph theory captures connectivity structure independent of any particular embedding, whether in physical space, social networks, or conceptual domains.

Abstraction Reveals Structure

The power lies in discarding irrelevant information. Bridges problem: distances, angles, physical appearance—all irrelevant. Only connectivity matters. This is mathematical abstraction—isolate structure from substance. Find the right level to analyze without distraction.

My polyhedron formula demonstrates topological invariance: $V - E + F = 2$, where $V$ is vertices, $E$ is edges, and $F$ is faces. This holds for any polyhedron homeomorphic to a sphere. Cube: 8 vertices, 12 edges, 6 faces (8 - 12 + 6 = 2). Tetrahedron: 4, 6, 4 (4 - 6 + 4 = 2). Properties preserved under continuous transformation—stretching allowed, tearing forbidden. Topology studies what remains invariant through deformation.

The complex plane provides another example. Representing complex numbers as $a + bi$ becomes clearer when visualized geometrically: horizontal axis shows real parts, vertical axis shows imaginary parts. Operations transform into movements: addition becomes vector addition, multiplication becomes rotation and scaling. My formula $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ reveals that exponential functions with imaginary inputs trace circles—exponential growth transforms into rotation through the imaginary unit. Geometry illuminates algebra.

Modern applications proliferate: Internet routing finds paths through networks, DNA sequencing reconstructs sequences from fragments via Eulerian paths through overlap graphs, social network analysis examines connectivity and influence spread, circuit design traces PCB layouts without crossing. Graph neural networks enable machine learning on relational data. Topological data analysis extracts structure from high-dimensional datasets.

Nothing takes place in the world whose meaning is not mathematical. Graph theory demonstrates this principle. By abstracting to essential structure—nodes and edges—we solve problems intractable through geometric calculation. Different geometric representations serve different purposes, but underlying connectivity remains fundamental.

Let us calculate. Mathematics is not abstract edifice but practical tool. When we recognize that cognitive maps, color spaces, and city bridges share common structure, we gain universal framework applicable across domains. This is versatility through integration—using methods from one area in another, trusting patterns that persist through abstraction.