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.

21786 questions
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Is Sage on the same level as Mathematica or Matlab for graph theory and graph visualization?

The context: I'm going to start working on a project that involves running predefined algorithms (and defining my own) for very big graphs (thousands of nodes). Visualization would also be welcome if possible. This is a research project and the goal…
50
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7 answers

What is the optimal path between $2$ fixed points around an invisible obstructing wall?

Every day you walk from point A to point B, which are $3$ miles apart. There is a $50$% chance each walk that there is an invisible wall somewhere strictly between the two points (never at A or B). The wall extends $1$ mile in each direction…
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What is difference between cycle, path and circuit in Graph Theory

I am currently studying Graph Theory and want to know the difference in between Path , Cycle and Circuit. I know the difference between Path and the cycle but What is the Circuit actually mean.
Sudeep Acharya
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46
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For which $n\in\Bbb N$ can we divide $\{1,2,3,...,3n\}$ into $n$ subsets each with $3$ elements such that in each subset $\{x,y,z\}$ we have $x+y=3z$?

For which $n\in \mathbb{N}$ can we divide the set $\{1,2,3,\ldots,3n\}$ into $n$ subsets each with $3$ elements such that in each subset $\{x,y,z\}$ we have $x+y=3z$? Since $x_i+y_i=3z_i$ for each subset $A_i=\{x_i,y_i,z_i\}$, we have $$4\sum…
46
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5 answers

Graph terminology: vertex, node, edge, arc

Precisely speaking, what is the difference between the graph terms of ("vertex" vs. "node") and ("edge" vs. "arc")? I have read that "node" and "arc" should be used when the graph is strictly a tree. If there is a precise rule or protocol, please…
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Graph theory software?

Is there any software that for drawing graphs (edges and nodes) that gives detailed maths data such as degree of each node, density of the graph and that can help with shortest path problem and with algorithms such as Dijkstra ? Thanks!
graphtheory92
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45
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Is it possible to have a spherical object with only hexagonal faces?

If so, what would be the most efficient algorithm for generating spheres with different number of hexagonal faces at whatever interval required to make them fit uniformly or how might you calculate how many hexagonal faces are required for each…
CoryG
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45
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What do the eigenvectors of an adjacency matrix tell us?

The principal eigenvector of the adjacency matrix of a graph gives us some notion of vertex centrality. What do the second, third, etc. eigenvectors tell us? Motivation: A standard information retrieval technique (LSI) uses a truncated SVD as a…
42
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6 answers

Four color theorem disproof?

My brother in law and I were discussing the four color theorem; neither of us are huge math geeks, but we both like a challenge, and tonight we were discussing the four color theorem and if there were a way to disprove it. After some time scribbling…
Doktor J
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Counting trails in a triangular grid

A triangular grid has $N$ vertices, labeled from 1 to $N$. Two vertices $i$ and $j$ are adjacent if and only if $|i-j|=1$ or $|i-j|=2$. See the figure below for the case $N = 7$. How many trails are there from $1$ to $N$ in this graph? A trail is…
Dave Radcliffe
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40
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4 answers

Belt Balancer problem (Factorio)

So this question is inspired by the following thread: https://forums.factorio.com/viewtopic.php?f=5&t=25008 In it, the poster is examining an $8$-belt balancer (more on that to come) which he shows fails to satisfy a desirable property, which he…
Justin Benfield
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40
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Difference between a sub graph and induced sub graph.

I have the following paragraph in my notes: If $G=(V,E)$ is a general graph . Let $U\subseteq V$ and let $F$ be a subset of $E$ such that the vertices of each edge in $F$ are in $U$ , then $H=(U,F)$ is also a general graph and $H$ is a subgraph…
patang
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39
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Show that there's a unique minimum spanning tree if all edges have different costs

Show that there's a unique minimum spanning tree (MST) in case the edges' weights are pairwise different $(w(e)\neq w(f) \text{ for } e\neq f)$. I thought that the proof can be done for example by contradiction, saying that we have $2$ different…
totpiko
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39
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Prove that at a party of $25$ people there is one person knows at least twelve people.

So, the full problem goes like this: There are $25$ people at a party. Assuming that among any three people, at least two of them know each other, prove that there exists one person who must know at least twelve people. I've been stuck on this…
37
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3 answers

Painting the plane, and finding points one unit apart

An old (rather easy) contest problem reads as follows: Each point in a plane is painted one of two colors. Prove that there exist two points exactly one unit apart that are the same color. This proof can be easily written by constructing an…