Consider the network shown in Figure 518 there is an edge be

Consider the network shown in Figure -5.18: there is an edge between each pair of nodes, with five of the edges corresponding to positive relationships, and the other five of the edges corresponding to negative relationships. Each edge in this network participates in three triangles: one formed by each of the additional nodes who is not already an endpoint of the edge. (For example, the A-B edge participates in a triangle on A, B, and C, a triangle on A, B, and D, and a triangle on A, B, and E. We can list triangles for the other edges in a similar way.) For each edge, how many of the triangles it participates in are balanced, and how many are unbalanced. (Notice that because of the symmetry- of the network, the answer will be the same for each positive edge, and also for each negative edge; so it is enough to consider this for one of the positive edges and one of the negative edges.)

Solution

The triangle formed by using all the positive or negative edges in it is called as balanced triangle.

The triangle formed by using both positive and negative edges is called as Unbalanced triangle.

For the edge AB the triangles formed are given as , ABC - Unbalanced triangle ABD - Unbalanaced triangle , ABE - unbalanced triangle .

For the edge AC , ACB - Unbalanced traingle , ACE - Unbalanced triangle , ACD - Unbalanced triangle.

For the edge CE , CEA - Unbalanced triangle, CEB - Unbalanaced triangle, CED- Unbalanced triangle.

For the edge DE , DEA - Unbalanced triangle, DEB- Unbalanced triangle, DEC - Unbalanced triangle.

For the edge BD , BDA - Unbalanced triangle , BDE - Unbalanced triangle, BDC - Unbalanced traingle.

For the edge AE , AEC - Unbalanced triangle, AED - Unbalanced triangle , AEB- Unbalanced triangle.

For the edge AD , ADE- Unbalanced triangle, ADB - Unbalanced triangle, ADC - Unbalanced triangle.

For the edge BC , BCD - Unbalnced triangle, BCE - Unbalanced triangle, BCA - Unbalanced triangle.

For the edge BE, BEA - Unbalanced triangle, BED - Unbalanced triangle , BEC- Unbalanced triangle

For the edge CD, CDA - Unbalanced triangle, CDB- Unbalanced triangle , CDE- Unbalanced triangle.