# Why is this the obvious? (graph Theory)

• Firepanda
In summary, the Perron Frobenius theorem states that for connected graphs, the eigenvalue of 1 is simple and has a strictly positive eigenvector. This relates to the problem of bipartite graphs, where changing the sign of every other entry in an eigenvector results in an eigenvector associated with negative the first eigenvalue.
Firepanda
http://www.win.tue.nl/~aeb/srgbk/node10.html

Under the 'For connected graphs all is clear from above'

I've been asked to prove this for connected graphs, but I don't see how this is so clear?

Can anyone help me? Thanks

Last edited by a moderator:
It says "it is clear from the above" but you have cut off the "above"! What did it say before that theorem?

HallsofIvy said:
It says "it is clear from the above" but you have cut off the "above"! What did it say before that theorem?

Well I googled my problem and found that link after some pages. I'm not sure what it said.

But it probably was something like the Perron Frobenius theorem.

From my notes I have (Perron Frobenius Thereom)

'Let A be in the set of all Schrodinger operators, and suppose a graph G is connected.

Then eigenvalue(1) is simple (i.e. has multiplicity 1) and its eigenspace is generated by an eigenvector that is strictly positive everywhere.'My general question that I was asked to do came from this email he sent me:

'These graphs are bipartite, meaning that if you change the sign of every other entry in an eigenvector then you get an eigenvector associated with negative the first eigenvalue (prove this!)'

## 1. Why is graph theory considered to be obvious?

Graph theory is considered to be obvious because its basic concepts and principles can be easily understood and applied to a wide range of real-world problems. The visual representation of graphs makes it easier for people to grasp the connections and relationships between different objects or entities.

## 2. How does graph theory help in problem-solving?

Graph theory provides a mathematical framework for modeling and analyzing complex problems that involve networks or systems. By breaking down a problem into smaller components and representing them as nodes and edges on a graph, graph theory allows for efficient problem-solving and optimization.

## 3. What are the key applications of graph theory?

Graph theory has a wide range of applications in various fields, including computer science, engineering, social sciences, and transportation. Some common applications include network analysis, route planning, social network analysis, and circuit design.

## 4. Can graph theory be applied to real-world problems?

Yes, graph theory is often used to solve real-world problems that involve connections or relationships between different objects or entities. Its applications are diverse and it has been successfully applied in various fields, such as transportation, biology, and finance.

## 5. Is graph theory a difficult subject to learn?

While graph theory may seem intimidating at first, its basic concepts and principles can be easily grasped with some practice. It is a fundamental branch of mathematics and with some dedication, anyone can learn and apply it to solve complex problems.

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