Mixing
Minimum number of edges formula?
$begingroup$
Is there a formula of finding the minimum number of edges?
I know that for the maximum number of edges it is
$(n-k)(n-k+1)/2$ where $n$ is the number of vertices and $k$ is number of components.
I was reading online and some said $(n-1)(n-2)/2 +1$
however in my case I know there are $n=10$ vertices and $k=3$ components, hence the max number of edges is $28$.
If I apply the formula for min. found online I get $37$ edges which is clearly wrong.
graph-theory
asked Jan 30 at 18:09
HidawHidaw
460624
$endgroup$
add a comment |
$begingroup$
Is there a formula of finding the minimum number of edges?
I know that for the maximum number of edges it is
$(n-k)(n-k+1)/2$ where $n$ is the number of vertices and $k$ is number of components.
I was reading online and some said $(n-1)(n-2)/2 +1$
however in my case I know there are $n=10$ vertices and $k=3$ components, hence the max number of edges is $28$.
If I apply the formula for min. found online I get $37$ edges which is clearly wrong.
graph-theory
asked Jan 30 at 18:09
HidawHidaw
460624
$endgroup$
2
$begingroup$
Clarify? A graph can have no edges... What are n and k?
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:11
1
$begingroup$
The minimum number of edges is $0$. If you know some other things about the graphs under consideration, they need to be included.
$endgroup$
– Greg Martin
Jan 30 at 18:20
$begingroup$
@GregMartin is a graph with zero edges considered simple graph? In my case I have 10vertices and 3 components is it correct to draw 5 vertices in one component, 3 vertices in the second component and 2 vertices in the third component?
$endgroup$
– Hidaw
Jan 30 at 18:21
add a comment |
$begingroup$
Is there a formula of finding the minimum number of edges?
I know that for the maximum number of edges it is
$(n-k)(n-k+1)/2$ where $n$ is the number of vertices and $k$ is number of components.
I was reading online and some said $(n-1)(n-2)/2 +1$
however in my case I know there are $n=10$ vertices and $k=3$ components, hence the max number of edges is $28$.
If I apply the formula for min. found online I get $37$ edges which is clearly wrong.
graph-theory
asked Jan 30 at 18:09
HidawHidaw
460624
$endgroup$
Is there a formula of finding the minimum number of edges?
I know that for the maximum number of edges it is
$(n-k)(n-k+1)/2$ where $n$ is the number of vertices and $k$ is number of components.
I was reading online and some said $(n-1)(n-2)/2 +1$
however in my case I know there are $n=10$ vertices and $k=3$ components, hence the max number of edges is $28$.
If I apply the formula for min. found online I get $37$ edges which is clearly wrong.
graph-theory
graph-theory
asked Jan 30 at 18:09
HidawHidaw
460624
asked Jan 30 at 18:09
HidawHidaw
460624
edited Jan 30 at 18:11
Hidaw
asked Jan 30 at 18:09
HidawHidaw
460624
asked Jan 30 at 18:09
HidawHidaw
460624
asked Jan 30 at 18:09
HidawHidaw
460624
460624
2
$begingroup$
Clarify? A graph can have no edges... What are n and k?
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:11
1
$begingroup$
The minimum number of edges is $0$. If you know some other things about the graphs under consideration, they need to be included.
$endgroup$
– Greg Martin
Jan 30 at 18:20
$begingroup$
@GregMartin is a graph with zero edges considered simple graph? In my case I have 10vertices and 3 components is it correct to draw 5 vertices in one component, 3 vertices in the second component and 2 vertices in the third component?
$endgroup$
– Hidaw
Jan 30 at 18:21
add a comment |
2
$begingroup$
Clarify? A graph can have no edges... What are n and k?
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:11
1
$begingroup$
The minimum number of edges is $0$. If you know some other things about the graphs under consideration, they need to be included.
$endgroup$
– Greg Martin
Jan 30 at 18:20
$begingroup$
@GregMartin is a graph with zero edges considered simple graph? In my case I have 10vertices and 3 components is it correct to draw 5 vertices in one component, 3 vertices in the second component and 2 vertices in the third component?
$endgroup$
– Hidaw
Jan 30 at 18:21
2
2
$begingroup$
Clarify? A graph can have no edges... What are n and k?
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:11
$begingroup$
Clarify? A graph can have no edges... What are n and k?
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:11
1
1
$begingroup$
The minimum number of edges is $0$. If you know some other things about the graphs under consideration, they need to be included.
$endgroup$
– Greg Martin
Jan 30 at 18:20
$begingroup$
The minimum number of edges is $0$. If you know some other things about the graphs under consideration, they need to be included.
$endgroup$
– Greg Martin
Jan 30 at 18:20
$begingroup$
@GregMartin is a graph with zero edges considered simple graph? In my case I have 10vertices and 3 components is it correct to draw 5 vertices in one component, 3 vertices in the second component and 2 vertices in the third component?
$endgroup$
– Hidaw
Jan 30 at 18:21
$begingroup$
@GregMartin is a graph with zero edges considered simple graph? In my case I have 10vertices and 3 components is it correct to draw 5 vertices in one component, 3 vertices in the second component and 2 vertices in the third component?
$endgroup$
– Hidaw
Jan 30 at 18:21
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
To make a single component with $m$ vertices, you need $m-1$ edges. In your case, you need to have three components with $a,b,c$ vertices respectively, with $a+b+c=10$. It doesn't matter what you choose for $a,b,c$, in any case you need $$(a-1) + (b-1) + (c-1) = 10-3=7$$ edges to make such a graph.
So in general the formula is just $n-k$.
answered Jan 30 at 18:31
Nathaniel MayerNathaniel Mayer
1,863516
$endgroup$
add a comment |
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1 Answer
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$begingroup$
To make a single component with $m$ vertices, you need $m-1$ edges. In your case, you need to have three components with $a,b,c$ vertices respectively, with $a+b+c=10$. It doesn't matter what you choose for $a,b,c$, in any case you need $$(a-1) + (b-1) + (c-1) = 10-3=7$$ edges to make such a graph.
So in general the formula is just $n-k$.
answered Jan 30 at 18:31
Nathaniel MayerNathaniel Mayer
1,863516
$endgroup$
add a comment |
$begingroup$
To make a single component with $m$ vertices, you need $m-1$ edges. In your case, you need to have three components with $a,b,c$ vertices respectively, with $a+b+c=10$. It doesn't matter what you choose for $a,b,c$, in any case you need $$(a-1) + (b-1) + (c-1) = 10-3=7$$ edges to make such a graph.
So in general the formula is just $n-k$.
answered Jan 30 at 18:31
Nathaniel MayerNathaniel Mayer
1,863516
$endgroup$
add a comment |
$begingroup$
To make a single component with $m$ vertices, you need $m-1$ edges. In your case, you need to have three components with $a,b,c$ vertices respectively, with $a+b+c=10$. It doesn't matter what you choose for $a,b,c$, in any case you need $$(a-1) + (b-1) + (c-1) = 10-3=7$$ edges to make such a graph.
So in general the formula is just $n-k$.
answered Jan 30 at 18:31
Nathaniel MayerNathaniel Mayer
1,863516
$endgroup$
To make a single component with $m$ vertices, you need $m-1$ edges. In your case, you need to have three components with $a,b,c$ vertices respectively, with $a+b+c=10$. It doesn't matter what you choose for $a,b,c$, in any case you need $$(a-1) + (b-1) + (c-1) = 10-3=7$$ edges to make such a graph.
So in general the formula is just $n-k$.
answered Jan 30 at 18:31
Nathaniel MayerNathaniel Mayer
1,863516
answered Jan 30 at 18:31
Nathaniel MayerNathaniel Mayer
1,863516
answered Jan 30 at 18:31
Nathaniel MayerNathaniel Mayer
1,863516
answered Jan 30 at 18:31
Nathaniel MayerNathaniel Mayer
1,863516
1,863516
add a comment |
add a comment |
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2
$begingroup$
Clarify? A graph can have no edges... What are n and k?
$endgroup$
– Nathaniel Mayer
Jan 30 at 18:11
1
$begingroup$
The minimum number of edges is $0$. If you know some other things about the graphs under consideration, they need to be included.
$endgroup$
– Greg Martin
Jan 30 at 18:20
$begingroup$
@GregMartin is a graph with zero edges considered simple graph? In my case I have 10vertices and 3 components is it correct to draw 5 vertices in one component, 3 vertices in the second component and 2 vertices in the third component?
$endgroup$
– Hidaw
Jan 30 at 18:21