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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Does there exists a growing sequence of simple connected regular graphs of girth $k$ ($k \geq 5$) with uniformly bounded diameter?

Fix any $k \geq 5$. Does there exist a growing sequence of simple connected REGULAR graphs, each of girth $k$, where the number of vertices $n$ goes to infinity and where the diameter of each graph is uniformly bounded in $n$?
Aftermath 12345
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Lower bound of crossing number (minimum) of K7

I am able to construct K7 with crossing number 9 from K6 with crossing number 3. Hence, I know the crossing number for K7 is at most 9; though, the number may not be sharp. Now, I heard it is possible to prove the crossing number of K7 is at least 7…
Sean
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Examples of problems/proofs which can be (surprisingly) represented in terms of graphs

I am looking for examples of problems or proofs in mathematics which have a equivalent representation in terms of graphs, which makes solving the problem easier. For example, the problem of finding how many inversions are required to sort an array…
Agile_Eagle
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What is the smallest diameter of a set of $n$ points in the plane which are all at least 2 meters apart from each other?

This question is similar to https://en.wikipedia.org/wiki/Circle_packing_in_a_circle except I am looking for the smallest diameter, i.e: I want the smallest maximum distance between the centers of the circles, rather than the smallest circle which…
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Shortest Distance between all pairs of nodes in graph

As the title says, how should I go about finding the shortest distance between all pairs of nodes (Each node has x and y co-odrinates associated with it) on a graph? A brute force method is to run shortest path finding algorithms between all the…
yasht
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Properties of prime sum graphs

The prime sum graph $P_n$ on the vertex set $V(P_n) = \{1,\dots, n\}$ has an edge $e = xy$ when $x+y$ is prime. It is easy to show that any such $P_n$ is bipartite (put odd numbers in one part and evens in the other). Thus, for a Hamilton cycle to…
PhysMath
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difference between "minimal" and "minimum" edge cuts.

I was going through the topic about connectivity of graphs. There it was mentioned about the terms "minimum edge cut" and "minimal edge cut". I know both are the sets of edges if removed from the graph $G$, makes $G$ disconnected. But I am unable to…
monalisa
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Knight tour problem??

Consider an n × n chess board. For what values of n is it possible to find a knight’s tour around the board which uses every possible move just once (in one direction or the other). Here on what factors does n depends?? Any Hints. Is every possible…
TLE
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How does this follow from the vanilla style duality of linear programming

The first principles of duality in LP state simply that if you have a primal problem of the following form Then i can write the dual LP automatically as: Let $\mathcal{C}$ be the set of cliques in G. Consider an arbitrary Now how do we write the…
user_1_1_1
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Random walk, Cat and mouse

Here is the problem. In graph G, on different vertices there is cat and mouse. Cat and mouse do independent random walk, but time is synchronous, in one unit of time both cat and mouse do one step. a)Estimate expected life duration of mouse, if cat…
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Isolated vertex probabilities for different random graphs

I'm trying to teach myself a little more on threshold probabilities for random graphs, and I'm looking at the probability that graphs have an isolated vertex, when we add on a few restrictions (when by a 'random graph' I mean we take the set of…
Warner B.
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Is there a use for undirected graphs defined by ternary relations?

We can define an undirected graph $G = (V,E)$ as a binary relation $R_2$ on $V$, where $x R_2 y$ and $y R_2 x$ when $\{x,y\}\in E$. When we draw these graphs, they seem almost unreasonably useful in visualising all sorts of data, and relationships…
Gus Kenny
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Projection between graphs extends to a covering space

This is exercise 1.A.10 on page 87 of Hatcher's book Algebraic topology. Let $X$ be the wedge sum of $n$ circles, with its natural graph structure, and let $\tilde X \to X$ be a covering space with $Y \subset \tilde X$ a finite connected…
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Mathematical notation for formulas involving trees

I am working on document that requires me to write such things as "$T_1$ is a descendant of $T_0$", or "$N_1$ is an parent of $N_2$". For now, I've been highjacking set notation for use in formulas, like $$ f(A) = \sum_{B\subset A}g(B) $$ where $B$…
ikh
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Ambiguity in the definition of graph homomorphism

Given graph $G$ and $H$ and a function between $f : G \rightarrow H$ between the vertex sets, we say that $f$ is a graph homomorphism iff for all vertexes $x$ and $y$ of $G$ such that $xy$ is an edge of $G$, it holds that $f(x)f(y)$ is an edge of…
goblin GONE
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