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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Prove a graph with degree $\ge \frac{n}{2}$ has diameter $\leq 2$

$G$ is an undirected connected graph that has an even number of nodes, and every node has $≥ n/2$ degree. Prove that it has diameter $≤ 2$. I understand that if two vertices are not connected, then each of them has at least $n/2$ edges…
ern36
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Minimum number of triangles in a graph

I got the following question during a Discrete Mathematics exam. I have absolutely no clue how to even start solving and I just wanted to share it since it looks like an interesting problem, perhaps someone is able to provide a nice proof :). Let…
Lupetto1927
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Proof that a strongly connected digraph has an irreducible adjacency matrix

I need to prove that a strongly connected digraph has an irreducible adjacency matrix. If anybody would be willing to give an advice on how to tackle this problem I would be thankful.
grap
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Can I use graph theory to prove this is the optimal way to connect dots?

I'm working on a project related to making broadband available in rural areas. As part of this project, I'm looking at the shortest distances between connected properties and those that are not connected. Consider the plot below: image show scatter…
dialectic
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A collection of sets that cover all edges in Kn?

The problem is the following: Let $\mathcal{F}$ be a family of distinct proper subsets of {1,2,...,n}. Suppose that for every $1\leq i\neq j\leq n$ there is a unique member of $\mathcal{F}$ that contains both $i$ and $j$. Prove that $\mathcal{F}$…
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Prove that a tree with a vertex $v$ of degree $k > 1$ has at least $k$ leaves

My assignment is the following: Prove that if a tree that has a vertex $v$ of degree $k > 1$ has at least $k$ leaves. $\delta_T(v)$ means the degree of $v$ in the tree $T$ This is the approach I have: Let there be a tree $T$. By statement we know…
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Smallest $k$-regular unit-distance graph

We can create arbitrary $k$-regular unit-distance graphs by using a "hyper-cube construction": taking a $k-1$-regular unit-distance graph, making a copy and translating it one unit distance that is not parallel to any of the existing edges, and…
qwr
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Average degree of graph from polyhedral complex

Consider a graph constructed as follows. We begin with a pure polyhedral complex $C$ of dimension $d$ in $\mathbb{R}^d$ -- for our purposes this is just a finite collection $\{P_1, \dots, P_k\}$ of distinct $d$-dimensional convex polytopes ("cells")…
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Show $\alpha(G) \leq 2$ implies G contains $K_{\left \lceil \frac{n}{3} \right \rceil}$ as a minor

G is simple. I want to show $\alpha(G) \leq 2$ implies G contains $K_{\left \lceil \frac{n}{3} \right \rceil}$ as a minor. This is what I have so far: If $\alpha(G) = 1$, the graph $G$ is complete and it must contain the desired minor. So we assume…
jonan
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Toys or Manipulatives for Exploring Graph Theory

I was talking recently with my daughters about non-planar graphs, like $K_{3,3}$, $K_5$, and the 7 bridges of Königsberg. They got pretty interested in it. Then we tried to dive into the Petersen graph and the concept of linkless embedding in 3…
LarsH
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Let $G$ a graph without a $P_4$ as induced subgraph. Prove that either $G$ or $\overline{G}$ is disconnected.

I've got the following problem: Let $G$ a graph not containing a $P_4$ (path with 4 nodes) a an induced subgraph. Prove, that either $G$ or $\overline{G}$ is disconnected. Initially, I assumed the above condition means that the graph does not…
StckXchnge-nub12
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$n$ people at a party and friendship

There are $n\geq 2$ people at a party. Each person has at least one friend inside the party. Show that it is possible to choose a group of no more than $\frac{n}{2}$ people at the party, such that any other person outside the group has a friend…
Behemooth
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Why should I care about Brooks' Theorem?

Brooks' Theorem says that If $G$ is any connected graph that's neither complete nor an odd cycle, then $\chi(G) \leq \Delta(G)$ (Where $\chi$ is the colouring number and $\Delta(G)$ is the maximum degree of $G$.) Why is this an important result?…
NNN
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Ramsey number $R(3,7)=23$

Im reading the prove the Ramsey of number $R(3,7)=23$ of "Some Graph Theoretic Results Associated with Ramsey’s Theorem"JACK and JAMES YACKEL. $G$ is an $(x,y)$-graph if $x>C(G)$ and $y>I(G)$, where $I(G)$ is the maximum number of points of $G$ that…
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Probability that graph with $6$ vertices and $5$ edges has a triangle.

How to calcultate a probability that a graph with $6$ vertices and $5$ edges has a triangle? So we have ${15\choose 5}=3003$ (labeled) graphs and ${6\choose 3} =20$ possible triangles. Let $m_i$ be a number of graphs with $i$-th triangle. Then…
nonuser
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