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Difficulty: HardTransportation Modes and Network Connectivity

In transport network analysis, topological connectivity for planar graphs can be evaluated using the Gamma index (γ\gamma), which compares the actual number of edges (ee) to the maximum possible number of edges (3(v−2)3(v - 2)) for a given number of vertices (vv). If a regional highway network connecting 55 urban centers consists of 77 direct road links, what is the Gamma index of connectivity for this network?

  1. 79\frac{7}{9} (or approximately 0.780.78)Answer
  2. B
    75\frac{7}{5} (or 1.401.40)
  3. C
    710\frac{7}{10} (or 0.700.70)
  4. D
    715\frac{7}{15} (or approximately 0.470.47)

Answer

The Gamma index of connectivity is 79\frac{7}{9} (or approximately 0.780.78).
The Gamma index (γ\gamma) measures the degree of connectivity of a planar network on a scale from 00 to 11 (or 0%0\% to 100%100\%). For a planar graph with 55 vertices, the maximum possible number of direct edges without route intersection outside vertices is 3(5−2)=93(5 - 2) = 9. Given 77 actual edges, the ratio is 79≈0.78\frac{7}{9} \approx 0.78.

Step-by-Step Solution

1
Identify the given network parameters
Number of vertices v=5v = 5 (urban centers) and number of edges e=7e = 7 (road links).
Topological network indices depend on counting nodes and direct links.
2
Calculate the maximum theoretical number of edges (emax⁡e_{\max}) for a planar graph
emax⁡=3(v−2)=3(5−2)=3×3=9e_{\max} = 3(v - 2) = 3(5 - 2) = 3 \times 3 = 9.
In planar transport network graphs (where routes cross only at nodes), the upper bound of edges is constrained by 3(v−2)3(v - 2).
3
Compute the Gamma index (γ\gamma)
γ=eemax⁡=79≈0.78\gamma = \frac{e}{e_{\max}} = \frac{7}{9} \approx 0.78 (or 77.8%77.8\%).
The Gamma index is defined as the ratio of actual edges to maximum theoretical edges.

Key Concept

Gamma Index of Network Connectivity
Estimated Time:2m 0s
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Topics

Transportation Modes and Network Connectivity

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