00:01
In this question, we need to make use of the adjacency matrices to answer the following question for the graph k3 and k4.
00:09
So first of all, i'll discuss about k3.
00:12
In this, we have been asked to find out number of path, number of paths of length and length in between two different vertices, between two.
00:39
Different vertices okay and n takes the value for n equals to one two three and four so let us see how many are there so first of all the total path of length end between two vertices will be given as follow so first of all what i am going to do first i will find out when n equals to one i will write down my adjacency matrices, adjacency matrices.
01:14
Okay, so what i will get? it is represented by a.
01:20
So i will get 011 -101 -1 -110 because if i draw, let's say we have this three vertices here and we have to draw this, this correct so let's say this is one two and three vertex so one is not related to one and it is related to two and three similarly for two and three i get this okay so you see this represent this is how we write out the adjacency matrices between one and three there is vortex so we write one and between one and one no vortex is there no edge is there so we write zero so this is how we get the adjacency matrix correct so this will be a raise to 1 when you get n is equals to 2 so in that case adjacency matrix will be a square so i have to just multiply 0111 -1 -1 -110 0 -11 1 -0 -1 -1 -0 -1 -0 1 -0 okay so multiplying this what i'm going to get is 2 -1 -1 -1 -1 -2 when n value is 3 i will get a cube i can write this as a square times a.
02:40
So a square i have already to 1 -1 -1 -1 -2 -1, then we have 1 -1 -2, and multiplied with a.
02:49
0 -11 -11 -1 -1 -0.
02:54
So this comes out to be equal to 233, 3 -2 -3 -3 -33.
03:05
Now, i have to find out for n equals to 4.
03:10
To 4 that will be a cube times a we already have a cube which is two three three three two three three two correct and a we have as zero at one one one one one one zero one one this is our a now what we can do similarly solving this i am going to get six five five five six five five five five now seeing this adjacency vector says what i can observe here is total path, here i will write down the total path for length one.
03:56
It is how much it is just one.
03:59
Then i will get the total path for length two.
04:07
Okay, so for two it is again one and then we have total path for length three so for length three from this adjacency matrices how many paths are there so there are three paths okay so again total path for length four from this matrices what we can observe is five from you see from the same path we have six sorry from the same vertices to the same vertices we have length as six but from one partices to other we have length as five.
04:50
So okay, similarly here we are having three.
04:53
So this is how we observed.
04:55
Now let's move to the next part for k4.
04:58
So for k4 if i draw, so k4 is my this.
05:03
K4 is what? it is 1, 2, 3 and 4.
05:07
If i label the vertices, so this is my k4...
