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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0answers
18 views

Recursion Simple question- help

Sir,Please help me in solving the recursion $F(n)=K_0\frac{F(n-1)}{n-1}+K_1\frac{F(n-2)}{n-2}$,preferably using power series for the values of $F(n)$ in-terms of n. $K_1$ and $K_2$ here are ...
0
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1answer
19 views

How would you incorporate probability into this graph theory problem?

A non-closed path is chosen at random on the complete graph K9. All paths are equally likely. What is the probability that the path contains the edges {23} and {34} given that it is length 6? Given ...
0
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1answer
16 views

Hamiltonian cycle adjacency sum Proof

Let $C$ be a Hamiltonian cycle on a graph with vertices labeled {$1,...,9$}. Prove that there are $3$ vertices adjacent in $C$ whose labels sum to at least $12$. I understand why this fact is true by ...
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1answer
22 views

Calculating the number of isomorphic classes of complete bipartite graph

How many isomorphism classes of complete bipartite graphs have exactly 10 vertices? I don't understand what the question is asking or how to go about solving it.
0
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0answers
26 views

Construction of a Huffman tree

My task is the following: Provide an example for the following: A complete Huffmann tree with $n=5, q=2$, lengths $l_1,......,l_5$ and $l_1>l_2>l_3>l_4$. (Draw the tree and give weights ...
1
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3answers
58 views

Recurrence solution of simple recurrence

Please help me to find the solution of the recurrence in terms of n(implies $(f(n))$ and also the summation of the recurrence up to infinity ($sum = \sum_{n=0}^\infty f(n)$) . ...
1
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1answer
23 views

Clustering analysis of a weighted graph

My data consists of a large weighted undirected graph of $n$ nodes. I need to group the nodes into $m$ clusters ($m < n$), such that nodes in a cluster are connected with heavy weights. What ...
1
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0answers
20 views

Maximal flow in flow-networks

I want to do the task (b),(c) and (d)in the picture above. I have done (b) correctly. For (c) I only found one (s-t) augmenting path, namely (s,1),(1,3),(3,2),(2,4),(4,t) and I only can push one ...
0
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1answer
12 views

Can the number of cycles in non-planar undirected graphs be computed in reasonable time?

Background: I'm experimenting with programs that create non-planar undirected graphs from three-dimensional meshes. A graph created by program A is not necessarily isomorphic to one created by program ...
0
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0answers
27 views

Miscellaneous questions about trees

I want to know which of the following claims are true: 1) Let T be a minimal spanning tree in G for a weight function w. Then T is also a minimal spanning tree for the weight function obtained from w ...
0
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1answer
18 views

Proof about spanning tress in graphs

Let $G=(V,E)$ be a graph and $T_i=(V,F_i),i=1,2$ two disjoint spanning trees in $G$. Let $f_1 \in F_1$. Prove that there is $f_2\in F_2 $ such that $T:=T_1-f_1+f_2$ is a spanning tree.
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2answers
22 views

Number of edge disjoint Hamiltonian cycles in a complete graph with even number of vertices.

In a complete graph with $n$ vertices there are $\frac{n−1}{2}$ edge-disjoint Hamiltonian cycles if $n$ is an odd number and $n\ge 3$. What if $n$ is an even number?
4
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1answer
55 views

Free product as automorphism group of graph

Let $A$ and $B$ be two groups. We define following graph $X$. The set of vertices is the left cosets $gA$ and $gB$ where $g\in A*B$ (By $A*B$, I mean the free product of $A$ and $B$). The edges of the ...
0
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0answers
9 views

(Alternative) Notation for set of successors of a vertex in a directed graph

I am looking for the standard notation for the set of successors/predecessors of a vertex of a directed graph. I have seen $N^{+}(v)$ and $N^{-}(v)$ used to represent the set of direct successors and ...
0
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0answers
16 views

Q: Finding probability of connection based on distance?

So, I am new to graph theory and statistics but have encountered a problem that I am not exactly sure how to solve. I have a graph with n nodes and am trying to determine the probability of connection ...
0
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0answers
31 views

Tree Traversal - Simple Puzzle type Issue.

This is a puzzle like question,based on Fibonacci like structure of the tree. Actually it is a short question with out any complex concepts. It appears bit big,since I have added explanations with ...
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0answers
12 views

Resizing Edge Weights Bases on Node Sizes

I am new to graph theory and I and trying to create edge weights based on the sizes of the connecting nodes. My problem in particular is as such: I have a directed graph of e-mails that were sent ...
3
votes
1answer
29 views

Programmatically recognizing symmetries of a polyhedron

I'm programming something, but I'm stuck at something which more math-oriented people probably can help me with. I am giving a polyhedron in the following form: for each vertex I get the cyclic order ...
0
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0answers
10 views

Can Wiener process on a fractal random graph be reduced to a levy flight?

Weiner process on small-world graphs is a Levy flight. But does the condition still hold for a random graph that connects the edges of a fractal?
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0answers
55 views

Find the flaw in my 1-page proof of the Four Color Theorem

The Four Color Theorem has been proven for quite a while now, so I'm not really breaking ground there. But last night, for some reason, it popped into my head and I started thinking about it. I feel I ...
1
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3answers
59 views

May directed graph be embedded into manifold?

May directed graph be embedded into manifold?How ?and what is the condition?
0
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1answer
46 views

Assigning $\pm 1$ values to the edges of a complete graph

I read this sentence in one combinatorics book. In graph $K_{100}$, there is a possible way to assigns number (value) from $\{+1,-1\}$ to each edge, so that the sum of all edge values connected to ...
1
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0answers
33 views

Prove that a graph $G$ that is isomorphic to its dual is not bipartite

Where $G$ is a simple connected graph and has $\ge 2$ vertices. I'm trying to understand the answer from Proving a graph is not bipartite but I don't understand this is true. ...
3
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1answer
170 views

Mathematician who talked about the probability of a “good” graph?

In my undergraduate years, one of my professors always talked about this one mathematician who was talking about "good" graphs and wondered about the existence of such a graph. Apparently this ...
1
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0answers
16 views

minimum number of unit distances required for a unit equilateral triangle

Problem. Suppose we have $n$ points on the plane. Among $\binom{n}{2}$ pairwise distances, there are $e$ number of unit distances. Find minimum $e$ ($e$ as a function of $n$) so that there is a ...
4
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0answers
42 views

Is a connected graph uniquely determined by its weighted 2-step graph?

This is an extension of a previous question: Is a graph uniquely determined by its weighted 2-step graph?. In that question I asked about arbitrary graphs; in this question I restrict to connected ...
0
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1answer
12 views

Simple question about indexing edges of an undirected graph.

As far as I understand, for an undirected graph $\mathcal{G}=(\mathcal{N},\mathcal{E})$, the set of edges is defined as unordered 2-element subsets of $\mathcal{N}$. So, for example, $\mathcal{E} = ...
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0answers
29 views

Probability that a subset of a degree-regular graph shares at least a certain number of mutual connections

Consider a set of $n$ vertices of common degree $p$. What is the probability that some subset of $x$ vertices from $n$ share $q$ mutual connections within that group of size $x$? i.e. If we have ...
5
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0answers
74 views
+100

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 ...
0
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1answer
29 views

Graph isomorphism problem for labeled graphs

In the case of unlabeled graphs, the graph isomorphism problem can be tackled by a number of algorithms which perform very well in practice. That is, although the worst case running time is ...
0
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2answers
36 views

Prove that if G is a tree in which all vertices have odd degree then G has odd size.

Prove that if G is a tree in which all vertices have odd degree then G has odd size. Good night, do not know how to approach this "prove". Can you give me tips to solve it?. Please.
4
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0answers
22 views

Largest Matching whose removal does not leave Eulerian components

Task: Given an undirected graph $G = (V, E)$, find a largest matching $M \subseteq E$ such that $G-M$ has no Eulerian components (i.e. all connected components of $G-M$ must have odd-degree ...
1
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0answers
25 views

directed simple graph, all paths from node $ v_0 $ to an other node $ v $, MATLAB

consider a directed simple graph $ G=(V,E) $ with $ V=\lbrace v_0,v_1,\ldots,v_k \rbrace $ and adjacency matrix $ A=(a_{ij}) $, where $ a_{ij}=1 $ means, that there is an arc from node $ v_i $ to node ...
0
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1answer
16 views

What is the difference between `Cross edge` and `Forward edge` in a DFS tree?

In the most general way, Let $G(V, E)$ be a graph, and $T(V', E')$ be the DFS tree of $G$. If an edge $(u, v) \in E'$ is neither a tree edge nor a back edge, How can we determine whether it's a ...
2
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1answer
38 views

Is a graph uniquely determined by its weighted 2-step graph?

Let $G$ be an undirected graph. Define the 2-step graph $G^{(2)}$ of $G$ to be the weighted graph whose vertices are the same as those of $G$ but whose edges correspond to 2-step paths in $G$. Thus ...
1
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0answers
22 views

Find all simple graphs with exactly one pair of vertices of the same degree. [closed]

A simple graph is a graph with no loops or double edges. Find all simple graphs with exactly one pair of vertices of the same degree.
7
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1answer
113 views
+50

Graph partition that span a third of edges

Given a graph G is easy to see that we have a partition $V=V_1 \cup V_2$ so that $$e(G[V_1])+e(G[V_2])\leq e(G)/2$$. How can we improve this result showing that we can choose $V_i$ such that ...
0
votes
2answers
26 views

How to find a pointset with unique distances

Is there a way to arrange N number of 2D points within a box so that the distances between the points are unique? I have an application where I can measure the distances between points with some ...
0
votes
0answers
33 views

Signing the attendance

Imagine N students sitting in a straight row, students are numbered 1 to N. Attendance sheet is first given to student 1, who uses his own pen to sign the sheet. Then he passes the sheet to student 2, ...
0
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1answer
38 views

Crossing edges at space

Let's say I have Graph $G(v, e)$ I want to draw the graph without crossing edges on space. By giving $(x, y, z)$ for any Vertex. How can I check if one edge crosses another?
0
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2answers
24 views

Proof d-regular graph has an equal number of vertices in its bipartition

Let $G$ be a $d$-regular graph. Suppose that $G$ is bipartite with bipartition $(A,B)$. Prove that if $d>0$ then $|A| = |B|$. Also why is this statement false when $d=0.$ I'm not sure how to show ...
2
votes
2answers
48 views

Graph containing every trees of size $n$ as subgraphs

What is the minimum number of edges of graph $G$, so that every tree of size $n$ is a subgraph of $G$? I personally managed to find a lower bound of $c n \log n $ and an upper bound of $C n \log ...
0
votes
0answers
21 views

Matrix norm to compare two graphs

I have the adjacency matrices of two undirected graphs. I want to measure how different the two matrices are in terms of the linkage. Both matrices have the same number of nodes, but they differ in ...
0
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1answer
30 views

Find a kernel in a directed graph.

It's a question from a sample exam I'm trying to solve but with no success yet. Let $G(V, E)$ be a directed graph. set $A \subseteq V$ is a kernel if: i. $\forall u,v\in A \implies (u, v), ...
0
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1answer
26 views

Finding all mapping between two isomorphic graphs

Is there any formula for counting all the mappings between two isomorphic graphs? I have the following two graphs. and I am trying to find the mappings in the following way. For each edge in ...
1
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1answer
14 views

Eccentricity of vertices in a graph when eccentricity of one vertex is given

I have a very basic doubt. If a vertex in any graph has the eccentricity two, then what can be concluded about eccentricities of other vertices in graph. Is the eccentricity of every vertex is less ...
0
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1answer
32 views

Removing cycle from the complete graph.

How can I remove $6-length$ cycle from the $K_6$ complete graph so that it'll result a $K_{3,3}$ bipartite graph? I've tried a couple of ways, but I can't get needed result. Maybe this decomposition ...
-1
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1answer
45 views

Is the following graph chordal? [on hold]

A very simple question: Is the following graph a chordal graph?
-1
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0answers
44 views

Count suggestions to be send

A site currently has N registered users. As in any social network two users can be friends. We wants the world to be as connected as possible, so we want to suggest friendship to some pairs of users. ...
-1
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1answer
47 views

Friends meeting at point

N friends live in different houses spread across the city.There are M roads connecting the houses. The road network formed is connected and does not contain self loops and multiple roads between same ...