A state transport planning authority in Nigeria is evaluating two infrastructure developme… — JAMB UTME

A state transport planning authority in Nigeria is evaluating two infrastructure development phases for a regional transport network. In Phase 1, the network consists of $8$ nodal settlements ( $v$ ) connected by $12$ direct highway corridors ( $e$ ). In Phase 2, the network expands by adding $2$ new settlement nodes and $6$ new highway corridors that integrate into the existing system. What is the net change in the network's Beta index ( $\beta$ ) of topological connectivity from Phase 1 to Phase 2?

An increase of $0.30$ Cevap
B
A decrease of $0.11$
C
An increase of $1.50$
D
A decrease of $0.90$

Cevap

The topological connectivity increases by

0.30

in the Beta index (

\beta

In graph theory applied to transport geography, the Beta index (

\beta

) measures the connectivity of a network graph by dividing the total number of edges (

e

) by the total number of vertices (

v

). In Phase 1,

\beta_1 = 12 / 8 = 1.50

. In Phase 2, the network accumulates

8 + 2 = 10

vertices and

12 + 6 = 18

edges, yielding

\beta_2 = 18 / 10 = 1.80

. The net change in connectivity is

1.80 - 1.50 = +0.30

, indicating a more connected network graph.

Adım Adım Çözüm

Calculate the Phase 1 Beta index

\beta_1 = \frac{e_1}{v_1} = \frac{12}{8} = 1.50

The Beta index evaluates topological network connectivity as the ratio of edges (

e

) to vertices (

v

Determine the cumulative vertices and edges for Phase 2

v_2 = 8 + 2 = 10

vertices;

e_2 = 12 + 6 = 18

edges

Phase 2 integrates newly constructed nodes and routes into the existing Phase 1 network framework.

Calculate the Phase 2 Beta index

\beta_2 = \frac{e_2}{v_2} = \frac{18}{10} = 1.80

The new network ratio is computed using the updated totals of edges and vertices.

Calculate the net change between Phase 1 and Phase 2

\Delta \beta = \beta_2 - \beta_1 = 1.80 - 1.50 = +0.30

Subtracting the Phase 1 index from the Phase 2 index provides the net increase in connectivity.

Anahtar Kavram

Topological Network Connectivity (Beta Index)

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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