Intereting Posts

Norm of a Kernel Operator
Characteristics method applied to the PDE $u_x^2 + u_y^2=u$
Finding $\lim\limits_{x\to0}x^2\ln (x)$ without L'Hospital
Why is $dy dx = r dr d \theta$
Are the $\mathcal{C}^k$ functions dense in either $\mathcal{L}^2$ or $\mathcal{L}^1$?
A function for which the Newton-Raphson method slowly converges?
$M \times N$ orientable if and only if $M, N$ orientable
The density — or otherwise — of $\{\{2^N\,\alpha\}:N\in\mathbb{N}\}$ for ALL irrational $\alpha$.
Legendre transform of a norm
G is group of order pq, pq are primes
Show that the Area of image = Area of object $\cdot |\det(T)|$? Where $T$ is a linear transformation from $R^2 \rightarrow R^2$
Orthonormal basis with specific norm
Why is the image of a smooth embedding f: N \rightarrow M an embedded submanifold?
Proof of Product Rule of Limits
showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$

Could anyone give a counter example for that theorem :

A graph G has exactly one vertex of degree $1$, then it contains a cycle.

I am so confused. I wonder that may I give a counter example which considers trees.

- Trees whose complement is also a tree
- Weak $k$-compositions with each part less than $j$
- Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?
- Discrete Math Bit String proof
- Correct way to calculate numeric derivative in discrete time?
- Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$

- Coloring Graph Problem
- What do the eigenvectors of an adjacency matrix tell us?
- Prove the identity Binomial Series
- $A^A$ in category of graphs
- A graph $G$ is bipartite if and only if $G$ can be coloured with 2 colours
- What are the applications of finite calculus
- Proving graph connectedness given the minimum degree of all vertices
- Proving component size based on number of edges and vertices.
- Connectedness of a regular graph and the multiplicity of its eigenvalue
- How to draw all nonisomorphic trees with n vertices?

The counter example does not exist.

Let $v_1$ be a vertex of $G$ with degree $1$. Then, let $v_2$ be its neighbor. Because the degree of $v_2$ is more than $1$, $v_2$ has a neighbor that is not $v_1$. Let that neighbor be $v_3$.

Now, $v_3$ has a neighbor that is not $v_2$ (because the degree of $v_3$ is at least $2$) and is not $v_1$ (because $v_1$ has degree $1$). Let that neighbor be $v_4$.

Now, $v_4$ has a neighbor that is not $v_3$ or $v_1$. If the neighbor is $v_2$, then $v_2,v_3,v_4$ is a cycle. Otherwise the neighbor is $v_5$ and repeat until a cycle is found.

If you allow *infinite* graphs, then the set of natural numbers $\mathbb{N}=\{0,1,2,3,\ldots\}$ with consecutive numbers considered adjacent is a counterexample: the vertex $0$ has degree $1$, while all others have degree $2$, but the graph contains no cycles.

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