How many triangles has a complete graph. Recursive Equation
$begingroup$
I need to find the solution with a recursive equation. I found the solution with combinations:
$${n}choose{}3$$
but I don't know how to find it with recursive equations. I don't find a pattern to resolve it
Thanks
discrete-mathematics graph-theory combinations
asked Dec 6 '18 at 20:36
emeeemee
43
$endgroup$
add a comment |
$begingroup$
I need to find the solution with a recursive equation. I found the solution with combinations:
$${n}choose{}3$$
but I don't know how to find it with recursive equations. I don't find a pattern to resolve it
Thanks
discrete-mathematics graph-theory combinations
asked Dec 6 '18 at 20:36
emeeemee
43
$endgroup$
add a comment |
$begingroup$
I need to find the solution with a recursive equation. I found the solution with combinations:
$${n}choose{}3$$
but I don't know how to find it with recursive equations. I don't find a pattern to resolve it
Thanks
discrete-mathematics graph-theory combinations
asked Dec 6 '18 at 20:36
emeeemee
43
$endgroup$
I need to find the solution with a recursive equation. I found the solution with combinations:
$${n}choose{}3$$
but I don't know how to find it with recursive equations. I don't find a pattern to resolve it
Thanks
discrete-mathematics graph-theory combinations
discrete-mathematics graph-theory combinations
asked Dec 6 '18 at 20:36
emeeemee
43
asked Dec 6 '18 at 20:36
emeeemee
43
asked Dec 6 '18 at 20:36
emeeemee
43
asked Dec 6 '18 at 20:36
emeeemee
43
asked Dec 6 '18 at 20:36
emeeemee
43
43
add a comment |
add a comment |
1 Answer
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$begingroup$
Let $T_n$ be the number of triangles in a complete graph of size $n$ for $ngeq 3$. Deduce that $T_{n+1} = T_n + binom{n}{2}$, as it amounts to considering if a triangle is contained in the complete subgraph on $n$ vertices or not. With $T_3=1$, it follows that $T_n=1+sum_{k=3}^{n-1}binom{k}{2}=1+sum_{k=3}^{n-1}k(k+1)/2$. Simplifying this will give you the answer.
answered Dec 6 '18 at 21:42
Alex R.Alex R.
24.9k12452
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1 Answer
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1 Answer
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$begingroup$
Let $T_n$ be the number of triangles in a complete graph of size $n$ for $ngeq 3$. Deduce that $T_{n+1} = T_n + binom{n}{2}$, as it amounts to considering if a triangle is contained in the complete subgraph on $n$ vertices or not. With $T_3=1$, it follows that $T_n=1+sum_{k=3}^{n-1}binom{k}{2}=1+sum_{k=3}^{n-1}k(k+1)/2$. Simplifying this will give you the answer.
answered Dec 6 '18 at 21:42
Alex R.Alex R.
24.9k12452
$endgroup$
add a comment |
$begingroup$
Let $T_n$ be the number of triangles in a complete graph of size $n$ for $ngeq 3$. Deduce that $T_{n+1} = T_n + binom{n}{2}$, as it amounts to considering if a triangle is contained in the complete subgraph on $n$ vertices or not. With $T_3=1$, it follows that $T_n=1+sum_{k=3}^{n-1}binom{k}{2}=1+sum_{k=3}^{n-1}k(k+1)/2$. Simplifying this will give you the answer.
answered Dec 6 '18 at 21:42
Alex R.Alex R.
24.9k12452
$endgroup$
add a comment |
$begingroup$
Let $T_n$ be the number of triangles in a complete graph of size $n$ for $ngeq 3$. Deduce that $T_{n+1} = T_n + binom{n}{2}$, as it amounts to considering if a triangle is contained in the complete subgraph on $n$ vertices or not. With $T_3=1$, it follows that $T_n=1+sum_{k=3}^{n-1}binom{k}{2}=1+sum_{k=3}^{n-1}k(k+1)/2$. Simplifying this will give you the answer.
answered Dec 6 '18 at 21:42
Alex R.Alex R.
24.9k12452
$endgroup$
Let $T_n$ be the number of triangles in a complete graph of size $n$ for $ngeq 3$. Deduce that $T_{n+1} = T_n + binom{n}{2}$, as it amounts to considering if a triangle is contained in the complete subgraph on $n$ vertices or not. With $T_3=1$, it follows that $T_n=1+sum_{k=3}^{n-1}binom{k}{2}=1+sum_{k=3}^{n-1}k(k+1)/2$. Simplifying this will give you the answer.
answered Dec 6 '18 at 21:42
Alex R.Alex R.
24.9k12452
answered Dec 6 '18 at 21:42
Alex R.Alex R.
24.9k12452
answered Dec 6 '18 at 21:42
Alex R.Alex R.
24.9k12452
answered Dec 6 '18 at 21:42
Alex R.Alex R.
24.9k12452
24.9k12452
add a comment |
add a comment |
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