Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [graph-theory]

Use this tag for questions in graph theory. Here a graph is a collection of vertices and connecting edges. Use (graphing-functions) instead if your question is about graphing or plotting functions.

Graph theory is the study of graphs, which is defined as an ordered pair $G = (V, E)$ comprising a set $V$ of vertices or nodes or points together with a set $E$ of edges or arcs or lines, which are 2-element subsets of V (i.e. an edge is associated with two vertices, and that association takes the form of the unordered pair comprising those two vertices).

Questions involve graph properties, graph algorithms, proofs and examples involving graphs, and applications of graph theory to other fields or practical ends.

Use instead for questions about graphing or plotting of functions.

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There are apparently $3072$ ways to draw this flower. But why?

This picture was in my friend's math book: Below the picture it says: There are $3072$ ways to draw this flower, starting from the center of the petals, without lifting the pen. I know it's based on combinatorics, but I don't know how to…
user265554
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Online tool for making graphs (vertices and edges)?

Anyone know of an online tool available for making graphs (as in graph theory - consisting of edges and vertices)? I have about 36 vertices and even more edges that I wish to draw. (why do I have so many? It's for pathing in a game) Only tool…
f20k
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What are good books to learn graph theory?

What are some of the best books on graph theory, particularly directed towards an upper division undergraduate student who has taken most the standard undergraduate courses? I'm learning graph theory as part of a combinatorics course, and would like…
Dom
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Mondrian Art Problem Upper Bound for defect

Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles? This is known as the Mondrian Art Problem. For…
Ed Pegg
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Number of simple edge-disjoint paths needed to cover a planar graph

Let $G=(V,E)$ be a graph with $|E|=m$ of a graph class $\mathcal{G}$. A path-cover $\mathcal{P}=\{P_1,\ldots,P_k\}$ is a partition of $E$ into edge-disjoint simple paths. The size of the cover is $\sigma(\mathcal{P})=k$. I am interested in upper and…
A.Schulz
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Graph theoretic proof: For six irrational numbers, there are three among them such that the sum of any two of them is irrational.

Problem. Let there be six irrational numbers. Prove that there exists three irrational numbers among them such that the sum of any two of those irrational numbers is also irrational. I have tried to prove it in the following way, but I am not…
Arpon Basu
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Motivation for spectral graph theory.

Why do we care about eigenvalues of graphs? Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so they must be important. I always assumed that…
Alexander Gruber
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Graph theory: adjacency vs incident

Okay, so I think if 2 vertices are adjacent to each other, they are incident to each other....or do I have it wrong? Is this just different terminology. I thought I was totally clear on this for my class, but now I am doubting myself reading the…
pqsk
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Why is the Traveling Salesperson Problem "Difficult"?

The Traveling Salesperson Problem is originally a mathematics/computer science optimization problem in which the goal is to determine a path to take between a group of cities such that you return to the starting city after visiting each city exactly…
stats_noob
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Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ Also, a few days ago, a friend of mine taught me that…
mathlove
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Proof a graph is bipartite if and only if it contains no odd cycles

How can we prove that a graph is bipartite if and only if all of its cycles have even order? Also, does this theorem have a common name? I found it in a maths Olympiad toolbox.
Asinomás
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How does TREE(3) grow to get so big? (Laymen explanation)

I am not a mathematician but I am interested in big numbers. I find them to be really interesting, almost god-like. I am watching a series of videos from David Metzler on YouTube. I have a basic understanding of some fast growing functions. David…
Josh Kerr
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What's the relation between topology and graph theory

I read the Wikipedia articles for both topology, graph theory (plus topological graph theory). Does topology encompass also graph theory? Or topology is only about studying shapes while graph theory is about relations and the two meet in topological…
Francesco
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How does the divisibility graphs work?

I came across this graphic method for checking divisibility by $7$. $\hskip1.5in$ Write down a number $n$. Start at the small white node at the bottom of the graph. For each digit $d$ in $n$, follow $d$ black arrows in a succession, and…
Quixotic
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Why a complete graph has $\frac{n(n-1)}{2}$ edges?

I'm studying graphs in algorithm and complexity, (but I'm not very good at math) as in title: Why a complete graph has $\frac{n(n-1)}{2}$ edges? And how this is related with combinatorics?
nkint
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