There are n 3 different particles p1 p2 pn arranged in a...

There are n >= 3 different particles p1, p2, . . . , pn arranged in a circle. If 1 < i < n then pi is between particles pi−1 and pi+1, while p1 is between pn and p2 (and pn is between pn−1 and p1). The temperature of each particle is the average of the temperatures of its two adjacent neighbors. Must every particle have the same temperature? Justify your answer.

Transcript

00:01 Hi here in the given question.

00:04 We are given that there are and people who shake hands with one another now, we need to prove that there must be two people there must be two people who shake hands with the same number of people.

00:53 So here we need to prove this using the graph theory principle.

01:04 So here in order to prove this we will start writing down the proof.

01:09 So here we need to prove that there are two people who has the same number of handshakes.

01:18 So here in our case, let us consider each person as a vertex of graph.

01:27 Each person is a vertex of a graph and edge between two vertex represents handshake and age represents handshake.

01:44 Now further as we are given that there are n people.

01:49 So here we have n vertices.

01:51 Now, let us assume that for the sake of contradiction, no two people shake hand with same number of people, which means here for each people.

02:33 We have unique number of handshakes from 0 to n minus 1.

02:47 So here we can say that the graph has a unique degree for each vertex.

03:01 So here now the degree of a vertex in a graph is defined as number of ages connected to it.

03:08 We know that degree of a graph is number of ages connected.

03:23 So here in context to this problem, the degree of vertex represent the number of handshakes.

03:29 So here in our case degree of vertex is equal to number of handshakes by the corresponding persons...

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