vagnhestagn

2022-10-08

Let $G$be an acyclic graph with $c$ components. Show that the number of edges of $G$ is $n-c$.

smh3402en

Beginner2022-10-09Added 11 answers

We can prove the result prove by using introduction on $n$,the number of vertices.The answer is obviously true for $n=1,2,3$. Let be the answer true for all trees with fewer than $n$ vertices. Let $T$ be a tree with $n$ vertices and let be $e$ an edge with end vertices $u$ and $v$. So the only path between $u$ and $v$ is $e$. Therefore deletion of $e$ from $T$ disconnects $T$. Now $T-e$ consists of exactly two components ${T}_{1}$ and ${T}_{2}$ say, and as there were no cycles tp begin with, each component is a tree. Let ${n}_{1}<n$ and ${n}_{2}<n$. Thus by induction hypothesis, number of edges in ${T}_{1}$ and ${T}_{2}$ are respectively ${n}_{1}-1$ and ${n}_{2}-1$. Hence the number of edges in $T={n}_{1}-1+{n}_{2}-1+1={n}_{1}+{n}_{2}-1=n-1$

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