A regional transport network connects 5 agricultural collection hubs (v = 5) using 7 prima… — JAMB UTME

A regional transport network connects $5$ agricultural collection hubs ( $v = 5$ ) using $7$ primary road segments ( $e = 7$ ) in a single connected network ( $p = 1$ ). What is the Alpha index ( $\alpha$ ) of this network graph?

$0.60$ Answer
B
$1.40$
C
$0.40$
D
$0.71$

Answer

The Alpha index of the transport network is

0.60

The correct answer is

0.60

. The Alpha index evaluates circuit redundancy in a network graph using the formula

\alpha = \frac{e - v + p}{2v - 5}

. Substituting

e = 7

v = 5

, and

p = 1

yields

\frac{7 - 5 + 1}{2(5) - 5} = \frac{3}{5} = 0.60

Step-by-Step Solution

Calculate the actual number of fundamental circuits (

u

) present in the network using

u = e - v + p

u = 7 - 5 + 1 = 3

circuits.

The cyclomatic number

u

measures the actual redundancy (number of closed loops) in a network graph.

Calculate the maximum possible number of circuits in a planar graph with

v = 5

using

2v - 5

Maximum circuits

= 2(5) - 5 = 5

In planar network topology, the maximum theoretical number of circuits for

v

vertices is

2v - 5

Calculate the Alpha index (

\alpha

) by dividing the actual circuits by the maximum possible circuits.

\alpha = \frac{u}{2v - 5} = \frac{3}{5} = 0.60

The Alpha index measures network connectivity as a ratio between

0

(tree network with no circuits) and

1

(fully connected planar network).

Key Concept

Alpha Index of Network Connectivity

Alternative Method

Calculate the cyclomatic number

u = 7 - 5 + 1 = 3

. Then express

u

as a percentage of the maximum possible planar circuits

2(5) - 5 = 5

, giving

3/5 = 60\% = 0.60

Estimated Time:1m 15s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Topics

More questions from this topic