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.

learn more… | top users | synonyms

7
votes
4answers
915 views

Returning Paths on Cubic Graphs Without Backtracking

I was once interested in the returning paths on cubic graphs . But I'm even more curious to have the number of ways without backtracking, which means doing one step forward and than one back (which ...
12
votes
3answers
6k views

How many connected graphs over V vertices and E edges?

Is there a way to calculate the number of simple connected graphs possible over given edges and vertices? Eg: 3 vertices and 2 edges will have 3 connected graphs But 3 vertices and 3 edges will have ...
15
votes
2answers
642 views

How many 2-edge-colourings of $K_n$ are there?

I'm writing a paper on Ramsey Theory and it would be interesting and useful to know the number of essentially different 2-edge-colourings of $K_n$ there are. By that I mean the number of essentially ...
42
votes
16answers
26k views

What are good books to learn graph theory?

What are some of the best books on graph theory, particularly directed towards an upper division undergraduate student who has taken most the standard undergraduate courses? I'm learning graph theory ...
42
votes
6answers
3k views

Motivation for spectral graph theory.

Why do we care about eigenvalues of graphs? Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so ...
1
vote
2answers
461 views

Could one be a friend of all?

The social network "ILM" has a lot of members. It is well known: If you choose any 4 members of the network, then one of these 4 members is a friend of the other 3. Proof: Is then among any 4 ...
8
votes
10answers
4k views

3 Utilities | 3 Houses puzzle?

There's a puzzle where you have 3 houses and 3 utilities. You must draw lines so that each house is connected to all three utilities, but the lines cannot overlap. However, I'm fairly sure that the ...
11
votes
2answers
260 views

How many directed graphs of size n are there where each vertex is the tail of exactly one edge?

In a research problem in an unrelated area, me and a student found it necessary to count the number of directed graphs with every vertex having one outward-pointing edge, with no restrictions on the ...
6
votes
4answers
14k views

Given a simple graph and its complement, prove that either of them is always connected.

I was tasked to prove that when given 2 graphs $G$ and $\bar{G}$ (complement), at least one of them is a always a connected graph. Well, I always post my attempt at solution, but here I'm totally ...
3
votes
1answer
4k views

How can I find the number of the shortest paths between two points on a 2D lattice grid?

How do you find the number of the shortest distances between two points on a grid where you can only move one unit up, down, left, or right? Is there a formula for this? Eg. The shortest path between ...
15
votes
5answers
13k views

How to calculate the number of possible connected simple graphs with $n$ labelled vertices

Suppose that we had a set of vertices labelled $1,2,\ldots,n$. There will several ways to connect vertices using edges. Assume that the graph is simple and connected. In what efficient (or if there ...
10
votes
2answers
4k views

How to construct a k-regular graph?

I have a hard time to find a way to construct a k-regular graph out of n vertices. There seems to be a lot of theoretical ...
8
votes
1answer
4k views

How to prove that a simple graph having 11 or more vertices or its complement is not planar?

It is an exercise on a book again.If a simple graph G has 11 or more vertices,then either G or is complement $\bar { G } $ is not planar. How to begin with this?Induction? Thanks for your help!
4
votes
1answer
410 views

Fastest way to try all passwords

Suppose you have a computer with a password of length $k$ in an alphabet of $n$ letters. You can write an arbitrarly long word and the computer will try all the subwords of $k$ consecutive letters. ...
3
votes
2answers
1k views

If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree?

Any help would be appreciated. If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree?
2
votes
1answer
1k views

Constructing a graph from a degree sequence

Let's say I'm given several degree sequences like {4,3,3,2,2} {3,3,3,3} {5,3,3,2,2,1} I can find the number of edges using the handshaking lemma But how do I construct a graph just given these ...
24
votes
13answers
11k views

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 ...
8
votes
2answers
7k views

Show that there's a 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 ...
15
votes
4answers
6k views

Why a complete graph has $\frac{n(n-1)}{2}$ edges?

I'm studying graphs in algorithm and complexity, (but I'm not very good at math) as in title: Why a complete graph has $\frac{n(n-1)}{2}$ edges? And how this is related with combinatorics?
10
votes
4answers
898 views

How many “good” graphs of size $n$ are there?

Let's a call a directed simple graph $G$ on $n$ labelled vertices good if every vertex has outdegree 1 and, when considered as if it were undirected, it is connected. How many good graphs of size $n$ ...
12
votes
4answers
34k views

Proving that the number of vertices of odd degree in any graph G is even

I'm having a bit of a trouble with the below question Given $G$ is an undirected graph, the degree of a vertex $v$, denoted by $\mathrm{deg}(v)$, in graph $G$ is the number of neighbors of $v$. ...
7
votes
1answer
426 views

How to get from Chebyshev to Ihara?

I have competing answers on my question about "Returning Paths on Cubic Graphs Without Backtracking". Assuming Chris is right the following should work. Up to one thing: The number of returning paths ...
3
votes
1answer
1k views

If a graph with $n$ vertices and $n$ edges there must a cycle?

How to prove this question? If a graph with $n$ vertices and $n$ edges it must contain a cycle?
10
votes
3answers
2k views

Is the empty graph connected?

Is the empty graph always connected ? I've looked through some sources (for example Diestels "Graph theory") and this special case seems to be ommited. What is the general opinion for this case ? As ...
4
votes
2answers
3k views

k-regular simple graph without 1-factor

Here's what I'm reading: every regular bipartite graph has a 1-factor. But I understand that not every regular graph has a 1-factor. So, I was thinking if it's possible to find a $k$-regular simple ...
2
votes
2answers
220 views

The use of any as opposed to every.

This is a really basic question, but it is one I never really thought about until now. Let $\mathscr{G}$ be a tree. Then every pair of vertices in $\mathscr{G}$ is connected by a unique walk. We ...
1
vote
2answers
6k views

Prove Petersen graph is not Hamiltonian using deduction and no fancy theorems

Prove Petersen graph is not Hamiltonian using basic terminology and deductions. I'm looking for an explanation without k-colouring or anything fancy like that since I haven't covered that in class. ...
0
votes
2answers
2k views

Help with graph induction question?

Given a graph $G$ with $n$ vertices, where $n$ is even, prove by induction that if every vertex has degree $n/2 + 1$, then $G$ must contain a 3-cycle. A 3-cycle is a set of 3 vertices, $a; b; c$ such ...
64
votes
16answers
74k views

Online tool for making graphs (vertices and edges)?

Anyone know of an online tool available for making graphs (as in graph theory - consisting of edges and vertices)? I have about 36 vertices and even more edges that I wish to draw. (why do I have so ...
37
votes
1answer
873 views

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ ...
24
votes
4answers
9k views

Logic question: Ant walking a cube

There is a cube and an ant is performing a random walk on the edges where it can select any of the 3 adjoining vertices with equal probability. What is the expected number of steps it needs till it ...
8
votes
2answers
3k views

Algorithm wanted: Enumerate all subsets of a set in order of increasing sums

I'm looking for an algorithm but I don't quite know how to implement it. More importantly, I don't know what to google for. Even worse, I'm not sure it can be done in polynomial time. Given a set of ...
13
votes
4answers
16k views

Proof a graph is bipartite if and only if it contains no odd cycles

How can we prove that a graph is bipartite if and only if all of its cycles have even order? Also, does this theorem have a common name? I found it in a maths Olympiad toolbox.
8
votes
1answer
925 views

Upper bound on $\chi(G)$ for a triangle-free graph

I'm struggling with the following question; For every graph $G$ such that $K_3 \not\subseteq G$ (i.e. $G$ does not contain a triangle), prove that $\chi(G) \leq 2\sqrt{n} +1$ (where $\chi(G)$ ...
0
votes
2answers
302 views

When a 0-1-matrix with exactly two 1’s on each column and on each row is non-degenerated? [1]

Let $A$ be an $n\times n$ matrix with entries in the set $\{0,1\}$ which has exactly two ones in each column and two ones in each row. Give necessary and sufficient conditions for the rank of $A$ to ...
10
votes
1answer
9k views

How many non-isomorphic graphs with n vertices and m edges are there?

Could someone tell me how to find the number of all non-isomorphic graphs with $m$ vertices and $n$ edges. (The graph is simple, undirected graph) In my particular problem, $m =20, n=180$ Attempt at ...
2
votes
2answers
2k views

Every $k$ vertices in an $k$ - connected graph are contained in a cycle.

Let $G$ be a $k$-connected graph. Meaning, $G$ has no less than $k$ vertices, and for every set of $k-1$ or less vertices, if we remove them from $G$, the graph stays connected (Of course, $G$ itself ...
6
votes
1answer
358 views

Known bounds and values for Ramsey Numbers

Is there a good online reference that lists known bounds on Ramsey numbers (and is relatively up to date)? The wikipedia page only has numbers for $R_2(n,m)$. I am specifically interested in known ...
5
votes
1answer
6k views

Powers of Adjacency Matrix (Determination of connection in Graph)

I am studying graph theories, and I am not sure about how power of adjacency works. I know $k$-th powers of A tells connection in graph, and I can read lengths between a vertex to vertex after taking ...
4
votes
2answers
932 views

Matrix graph and irreducibility

How do I prove that if $A\in\mathbb C^{n\times n}$ is a matrix then it is irreducible if and only if its associated graph (defined as at Graph of a matrix) is strongly connected? Update: Seeing as ...
3
votes
1answer
365 views

Ramsey Number Inequality: $R(\underbrace{3,3,…,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,…3}_k)-1)+2$

I want to prove that: $$R(\underbrace{3,3,...,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,...3}_k)-1)+2$$ where R is a Ramsey number. In the LHS, there are $k+1$ $3$'s, and in the RHS, there are $k$ ...
3
votes
1answer
174 views

$K_6$ contains at least two monochromatic $K_3$ graphs.

Let $K_n$ be a complete $n$ graph with a color set $c$ with $c=\{\text{Red}, \text{Blue}\}$. Every edge of the complete $n$ graph is colored either $\text{Red}$ or $\text{Blue}$. Since $R(3, 3)=6$, ...
2
votes
2answers
765 views

Chromatic polynomial of a grid graph

I have the following graph with $nm$ vertices: ...
2
votes
1answer
239 views

associativity in graph theory

Can anybody help me in clearing the facts how the associativity was proved in cartesian product of 3 graphs, and thus showing isomorphism. I can easily solve for the case when its two graphs. Taking ...
2
votes
1answer
567 views

Proving a simple graph is a connected graph

Does any proof exist that a simple graph with $n$ vertices such that the least vertex degree is $\geq \frac{n-1}{2}$ is a connected graph? (i.e. does such a proof have a name?)
1
vote
1answer
59 views

Chess King Tour 8x8 problem

Two squares on a chessboard are said to be neighbours if they have an edge or a corner on the board in common. This means that squares on the edge have 5 neighbours, on the corner have 3 neigbours, ...
1
vote
1answer
133 views

permutation group and cycle index question regarding peterson graph

A symmetry of graph $X$ is a permutation of the vertices that also happens to be a permutation of the induced edges. In particular, the distances between vertices are preserved by a symmetry. Show ...
47
votes
7answers
1k views

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). ...
43
votes
4answers
2k views

How does the divisibility graphs work?

I came across this graphic method for checking divisibility by $7$. $\hskip1.5in$ Write down a number $n$. Start at the small white node at the bottom of the graph. For each digit $d$ in ...
14
votes
2answers
6k views

Understanding the properties and use of the Laplacian matrix (and its norm)

I am reading the wikipedia article on the Laplacian matrix: http://en.wikipedia.org/wiki/Laplacian_matrix I don't understand what is the particular use of it; having the diagonals as the degree and ...