# Tagged Questions

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.

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### What is that thing that keeps showing in papers on different fields?

A few months ago, when I was studying strategies for the evaluation of functional programs, I found that the optimal algorithm uses something called Interaction Combinators, a graph system based on a ...
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### What can we say about the graph when many eigenvalues of the Laplacian are equal to 1?

The Laplacian of the graph has all the eigenvalues real and non-negative, the smallest being 0. I have a graph where the second smallest eigenvalue (the so called algebraic connectivity) is equal to 1....
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### Genus of the graph $K_{4,2,2,2}$.

What is the genus of the complete $4-$partite graph $K_{4,2,2,2}$? What i know: Since $K_{4,4,2}$ is a subgraph of $K_{4,2,2,2}$, and genus of $K_{4,4,2}$ is 2, $K_{4,2,2,2}$ has genus greater than ...
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### Minimal time gossip problem

The gossip problem (telephone problem) is an information dissemination problem where each of $n$ nodes of a communication network has a unique piece of information that must be transmitted to all the ...
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### “Semidirect product” of graphs?

The first subquestion is "has a standard notion of semidirect product been defined in graph theory"? If yes, i'd like to know if the definition i'm gonna give is equivalent to the standard one. I'd ...
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### Triangle dissection, no shared edges

It's possible to divide a triangle into smaller triangles such that no edge lengths are shared. Alternately, no two internal triangles share two vertices. The top three are the known simplest ...
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### Quotient Groups and Covering Spaces in Painting Hanging

Consider the $1$-out-of-$n$ painting hanging problem: Given $n$ nails in a wall, how can we hang a painting such that upon removal of any nail, it falls. This has a nice interpretation as a problem in ...
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### Who has a winning strategy in the hamilton-circle-game?

The game starts with a graph with $n$ vertices and no edges. The players alternately add edges until the graph contains a hamilton-circle. The player who made the last move loses. Who has a winning ...
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### Reconstruction Conjecture and Partial 2-trees

Reconstruction conjecture says that graphs (with at least three vertices) are determined uniquely by their vertex deleted subgraphs. This conjecture is five decades old. Searching relevant literature,...
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### Number of sets of vertices whose union of neighbours contains exactly $k$ vertices

Suppose a bipartite graph $g$ consisting of $2n(n-1),n\in\Bbb N,n>1$ vertices, is divided equally into two colors: red and blue, and is constructed as follows: For example, $g$ for $n=3$: If ...
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### Branching processes and couplings

A text I am reading is discussing ways to couple branching processes, and describes the following 2 pairings, the latter of which I am failing to understand. (I include the former for the sake of ...
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### Probability of the existence of a path of a specified length between any tw0 vertices in a random graph

Let $G$ be a graph with $n$ vertices, whose average degree is $k$. What is the probability that between any two vertices, there exists a path of length at most $l$? NOTE: For the above problem the ...
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### What is the Probability of Transmission Between Two Nodes in a Neural Network?

I have a network which is an Erdős–Rényi graph. It is a simple neural network with degree 0.7N where N is the number of nodes. Each weight between neurons is 1/N, meaning that if node n has fired ...
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### How is graph theory used to solve problems in number theory?

What are some applications of graph theory in number theory? How can a graph theory approach be useful to solving number theory problems? In general, is graph theory ever useful in making number ...
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### Given $100$ coplanar points, no $3$ collinear, then at most $70$ percent triangles formed using these points are acute-angled

(IMO-$1970$) Given $100$ coplanar points, no $3$ collinear, prove that at most $70$ percent of the triangles formed using these points are acute-angled. I know that one solution proceeds by showing ...
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### graph product that commutes with automorphism, and semi direct

Is there a way to construct a "product" of graphs $G\rtimes H$ such that $Aut(G \rtimes H ) \simeq Aut(G) \rtimes Aut(H)$? A related topic is "Semidirect product" of graphs? but not quite ...
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### A game may related to graph theory or topology

Last Sunday, I played a game with a group of people. The game is as follows: A group of people form a circle as shown below: Each person must remember how he/she is linked with his two neighbours. ...
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### When can an edge subset of a graph be extended to become an element of its cycle space?

Let $F$ be a set of edges in a graph $G$. Show that $F$ extends to an element of the cycle space of $G$ iff $F$ contains no odd cut. The context for this exercise is the following: Let $G = (V,E)$ be ...
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### Every tree has two leaves. Is my proof ok?

A tree is a connected acyclic graph. A leaf is a vertex of degree one. The distance $d(u,v)$ between two vertices $u$ and $v$ of a graph is the length of the shortest path from $u$ to $v$. Theorem. ...
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### Maximum number of cyclic quadruplets in tournament

Consider a tournament with $n$ contestants - that is, a complete graph directed graph $K_n$ where each edge is pointed one way or the other. We call a subset $\{a,b,c\}$ a "cyclic triplet" if each of ...
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### Traveling salesman problem: can a terrible strategy beat a good one?

Until yesterday, I was under the naive impression that constructing a weighted graph where the nearest-neighbour algorithm gives the worst possible route, would have the property that any other ...
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### Have these (extremely simple) classes of algebraic structures been considered in the literature? If so, what are they called?

Questions. Have the following kinds algebraic structures been considered in the abstract algebra literature etc.? If so, what are they really called? (I have used made-up terminology for the sake of ...
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### Parity of sum of Kronecker deltas in a graph

For some fixed $n\in\mathbb N$ let $G$ be a graph on the vertex set $\{1,\dots,n\}$ with a total number of $k$ edges $e_1,\dots, e_k$. For any vertex colouring $c(i)$ of the graph, $\delta_e$ is ...
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### Shortest closed loop containing all extreme points of a convex set

Suppose $S\subset \mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to\mathbb{R}^2$ is a continuous map with $\Gamma(0)=\Gamma(1)$. Suppose $\Gamma$ passes through all extreme points of $S$. (...
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### Constructing a directed graph from its spectrum

This is related to the following question from cs theory stack exchange: http://cstheory.stackexchange.com/questions/3742/reverse-graph-spectra-problem So it seems as if given a sequence of real ...
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### Slices of a hypercube

Take the unit $d$-cube with vertices $\{0,1\}^d$, and restrict to the vertices that lie between (or within one of) a pair of parallel hyperplanes. These vertices form a graph whose edges are the edges ...
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### Combinatorial vs. geometric symmetries of graphs and their drawings

Associated with a graph $G$ and its automorphism group $\text{Aut}(G)$ (reflecting its combinatorial symmetries) are drawings in the plane with - eventually - one or more (geometric) symmetry groups. ...
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### Unit Distance structure of Hoffman Singleton graph

This question has been bugging me since last 3 years. Prove or disprove that Hoffman Singleton is an unit distance graph in $\mathbb R^2$. For those who are new to unit distance graphs, A graph is ...
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### Ramsey (graph) theory question with tree and girth

Sorry for the abundance of questions I'm asking. Test is soon... Prove that for every tree $T$ and every $g \in \mathbb{N}$, exist $G$ with girth $g$, so that in any 2-edge-coloring of $G$ there is a ...
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### Edge Coloring of Kneser Graphs

Kneser graphs KG(n, k) are well known: vertices are all k-subsets of {1,2,...,n} with two sets connected iff they are disjoint. If the graph is odd (i.e., has an odd number of vertices) it is easy to ...
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### What do we know about graph degree sequences?

The sequence of sizes of single vertex cuts of a graph is called its degree sequence. Is there an agreed-upon name for the sequence of sizes of k-vertex cuts? What can be said about two graphs which ...
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### Determining graph minors quickly

I know very little graph theory. I am trying to determine if one graph, A, has as a minor another graph, B. I know the problem is slow in general, so I am looking for things that I might be able to ...
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### Points with power distances

It's possible for seven points to be at integral distances. I'm dallying with powered triangles, though, so I'm looking for point sets where all distances are powers of a fixed $x$ value. For example,...
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### Multi modal transport network and graph theory

I am attempting to model a multi layered transport graph with points that allow for travellers to laterally transfer between graphs, in order to make use of different transport nodes. Conceptually, ...
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### Single loop polyhedra

The odd antiprisms are both Eulerian and polyhedral, with the first implying that the edge can be represented with a single closed path. The Cuboctahedron also has that property. With the rule to ...
### Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$…
Looking at the images below, you recognize that the adajency matrix of the graph $A_G$ splits up into three different color submatrices, with $A_G=A_d+A_b+A_d$ (where $d$ is dark, damn...). It's ...