1 Another look at Harary graphs (15 points) The Harary graph...

1 Another look at Harary graphs (15 points) The Harary graph $H_{10,3}$ (discussed in lecture) is defined by arranging its 10 vertices in a circle, and adding an edge between vertices that are either next to or opposite each other on the circle. (5 points) Draw a diagram of $H_{10,3}$ using the vertices provided below. practice-exam1.pdf (5 points) Pick a vertex $v$ of the graph, and label all vertices with their distance from $v$. What is the diameter of $G$? (5 points) Is the graph $H_{10,3}$ bipartite? Give a reason why or why not.

Transcript

00:01 Hi, in this question we have to draw all the non -isomorphic graphs of order 5.

00:08 Now order 5 means the vertices should be 5 and size 5.

00:14 Now size 5 means the edges should be 5.

00:19 And thus the graph will be.

00:22 So thus these are the pairs of non -isomorphic graphs.

00:26 In the second part now we have been asked how many non -isomorphic graphs when have the degree 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 and 6 so let us see so we have to check if the graph is actually possible or if it is not possible we have considered the sequence degree sequence which has been given to us now from that what we have to do is we have to remove the first vortex if we remove this vortex that means the ages which will be there across means there it will will be having total six ages as the degree is six means there will be six ages if we remove that vertex having degree six that means the six ages will get removed so the degree of the another vertex will reduced right so of this two vertex is not reducing so it is as it is then we keep on going again we have to then write the numbers in an particular order so we have written it in an ascending order then again remove this vertex once removed the degree will get changed again write it in the ascending order remove this vertex once removed we will be getting the degrees will be like reducing so up doing moving forward in the same process we get this vertex so that means we come to know and this vertex is easy and it is possible to draw so does we can conclude from this that there is only a single unique graph which is possible with this degree sequence and therefore concluding that there is only one unique graph so does now this for the next part we have to draw six non isomorphic graph so does these are one two three four five and six non isomorphic graph where this is a circular part which we can see these are the loops loops means it consists of degree two so as as we can see in the first part 1, 2 and 3, so this vertex has 3 degree, this is 3 degree, this 1, this 1, this is 2 degree, and 2 degree...

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