Maths&physics
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#1
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can someone explain 4 a to me please?

what is meant by: the sum of the orders of all the nodes?

thank you
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Prasiortle
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#2
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(Original post by Maths&physics)
can someone explain 4 a to me please?

what is meant by: the sum of the orders of all the nodes?

thank you
A node is just another word for a vertex of the graph. The order (or degree) of a node is the number of edges that pass through that node.
The sum of the orders is just that: you write down the order of each node and sum them up.
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Maths&physics
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(Original post by Prasiortle)
A node is just another word for a vertex of the graph. The order (or degree) of a node is the number of edges that pass through that node.
The sum of the orders is just that: you write down the order of each node and sum them up.
I thought an order what the number of nodes?

I thought the degree/valency what the number of arcs attached to a node/vertex?
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Prasiortle
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#4
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(Original post by Maths&physics)
I thought an order what the number of nodes?

I thought the degree/valency what the number of arcs attached to a node/vertex?
The order of a graph is the number of nodes. You can also refer to the order of a node. It means exactly the same thing as the degree. "Number of arcs attached to" is exactly the same as "number of edges passing through". They're just different ways of phrasing it.
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Maths&physics
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(Original post by Prasiortle)
The order of a graph is the number of nodes. You can also refer to the order of a node. It means exactly the same thing as the degree. "Number of arcs attached to" is exactly the same as "number of edges passing through". They're just different ways of phrasing it.
ok, so what do they mean by: "Which implies that the sum of the orders of all the nodes is even"

sum of the orders (number of arcs connected to a node) is even because it is twice the numbers of arcs and anything multiplied by 2 is even.

"and therefore there must be an even (or zero) number of vertices of odd order hence there cannot be an odd number of vertices of odd order."

I dont understand the last bit?
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(Original post by Maths&physics)
ok, so what do they mean by: "Which implies that the sum of the orders of all the nodes is even"

sum of the orders (number of arcs connected to a node) is even because it is twice the numbers of arcs and anything multiplied by 2 is even.

"and therefore there must be an even (or zero) number of vertices of odd order hence there cannot be an odd number of vertices of odd order."

I dont understand the last bit?
Let's say there's an odd number of vertices of odd order. Then the sum of their orders is the sum of an odd number of odd numbers, which will be odd (as odd *odd = odd). Then we add on the orders of the even vertices, so we get odd + a bunch of evens, giving odd. But the total of the vertices' orders can't be odd. Thus there can't be an odd number of vertices of odd order.
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