**-1**

votes

**0**answers

28 views

### Random Spanning Tree Edge Probability [on hold]

I am working on a problem with a Loop Erased Random Walk used to create random spanning trees. The problem has many parts, but there are two hints to help with the more complicated problems
Figure ...

**0**

votes

**0**answers

46 views

### Are there any special properties of graph eigenvalues of perfect matchings?

Say if the graph is just a perfect matching of its vertices OR if it is an union of a few perfect matchings OR its an union of a few perfect matchings and a part of another?
Anything if one further ...

**-2**

votes

**0**answers

61 views

### What is the characterization of a graph Laplacian? [on hold]

Given a matrix, what properties must it have so that its ensured that there exists a graph whose Laplacian it would be? (...may be you can consider weighted and unweighted cases separately...)
And ...

**7**

votes

**0**answers

135 views

### Could there be an exact formula for the Ramsey numbers?

Let $R(k)$ denote the diagonal Ramsey number, i.e. the minimal $n$ such that every red-blue colouring of the edges of $K_n$ produces at least one monochromatic $K_k$.
Is it possible that there ...

**3**

votes

**1**answer

68 views

### Strongly asymmetric graphs

Asymmetric graphs are graphs that have a trivial automorphism group $\textrm{Aut}(G)$, i.e. the only graph isomorphism from $G$ to itself is the identity.
Let's call a graph $G$ strongly asymmetric ...

**0**

votes

**0**answers

54 views

### Number of graphs with M edges that does not contain K-clique [on hold]

If we consider the space of graphs $G(n,M)$ where $M$ denotes the number of edges. Is there any known way of calculating the number of graphs within this space that does not contain any k-cliques? Can ...

**5**

votes

**1**answer

78 views

### Isomorphic Hadwiger graphs

Let $G$ be a graph, then we define its Hadwiger graph $\textrm{Hadw}(G)$ in the following way:
$V(\textrm{Hadw}(G)) = \{S\subseteq (V(G): S\neq \emptyset\textrm{ and } S \textrm{ is connected}\}$;
...

**2**

votes

**1**answer

178 views

### Counting Ribbon graphs

Let $G$ be a ribbon graph (sometimes called fat graph) with $v$ vertices and $e$ edges. Furthermore each vertex is of degree $d$.
Q) What is the number of $G$ with the above properties? I mean does ...

**2**

votes

**0**answers

41 views

### relationship of max-sat and min-cut in theory and practice

I have been using MAX-SAT solver to obtain the exact ground state of ising spin glass model:
For 1D periodic model, for systems with 50 binary variables and interaction range of 15th nearest ...

**0**

votes

**1**answer

75 views

### Finding nodes with a particular weight in a graph

Say that an edge $e$ is incident to a node $v$ if one of its two extremes is $v$.
Then we can also say that $v$ is hit by $e$. We might define the notion of "weight of a node $v$" as the sum of all ...

**3**

votes

**1**answer

148 views

### Determinant of the oriented adjacency matrix of a tree

Let $(V,E)$ be a finite oriented directed graph, with vertices and edges ordered, and $M$ the $|V|\times |E|$ matrix with entries
$$ m_{ve} = \begin{cases} 1 &\text{if $e$ points at $v$}\\
-1 ...

**2**

votes

**3**answers

537 views

### Koebe–Andreev–Thurston theorem - where can I find a proof?

Koebe–Andreev–Thurston theorem (known also as the circle packing theorem) says that any planar graph can be realized by a set of (interior-) disjoint disks corresponding to vertices, such that two ...

**1**

vote

**0**answers

36 views

### Solutions for n such that the path number to the n-th node is also n in a complete, rooted, ordered k-ary tree

The path to find the $n$-th node within a complete, rooted, ordered $k$-ary tree can be represented by a number to the base $(k+1)$ that contains no zero digits. See A245905 in OEIS as an example for ...

**5**

votes

**1**answer

123 views

### Number of median graphs?

What is the number of $n$-vertex median graphs? These graphs generalize hypercubes and trees, and have many applications. It seems unlikely that a closed form expression is known, so I would also be ...

**-2**

votes

**1**answer

77 views

### how to reduce 3-colorable graph to this? [closed]

suppose we have a finite set X and a set S of subsets of X and we want to determine is there a subset S' of S such that all members of X belong to exactly one set in S' I think the best problem to ...

**0**

votes

**0**answers

11 views

### matching Robinson-Foulds distance and way to compute RF dist in Phylip

In Comparison of Phylogenetic Trees, Robinson D.F. and Foulds L.R., didn't show how to compute the RF distance between trees, counting the different partition generated by the removing of an internal ...

**-2**

votes

**0**answers

42 views

### when a given graph is 3-colorable? [closed]

I want to use graph 3-colorability to prove a problem is NP-complete But I'm not sure when a given graph is 3-colorable.
I think if it doesn't have any node to be connected to all 3 vertices of a ...

**-4**

votes

**0**answers

62 views

### About diagonal entries of the graph Laplacian [on hold]

[..in the following you can assume its a regular graph if necessary..]
Is anything special known about them?
Are they characterized in any other way?
Is the largest diagonal entry in any power of ...

**0**

votes

**0**answers

75 views

### independent subset problems [on hold]

I'm interested in the following which i suspect is probably a well studied problem.
Given a set $N=\{1,2,...,n\}$ and $M=\{1,2,...,m\}$ consider a map $$f:N\rightarrow 2^{M}$$ (elements of $N$ to ...

**1**

vote

**1**answer

65 views

### What is “graph-directed iterated function”?

Im translating an article about Rauzy fractal and I ran into this sentence:
...

**0**

votes

**0**answers

44 views

### $\Omega(G)$ is homeomorphism to $\Omega(T)$ [closed]

Is it possible that the end set of graph $G$ is homeomorphism to the set of end for every spanning tree of $G$ where $G$ has a cycle. In other hand, for given graph $G$ and spanning tree $T$ of $G$. ...

**9**

votes

**2**answers

376 views

### Can a parent and child node have the same type in a well-founded digraph tree?

$\newcommand\toward{\rightharpoonup}$It would help me to
understand something in a current research project if someone
could provide an example of directed graph $\langle
G,\toward\rangle$ with the ...

**7**

votes

**4**answers

203 views

### 4-regular graph with every edge lying in a unique 4-cycle

What are all 4-regular graphs such that every edge in the graph lies in a unique-4 cycle?
Among all such graphs, if we impose a further restriction that any two 4-cycles in the graph have at most one ...

**0**

votes

**1**answer

30 views

### Petersen 2-factor decomposition theorem for directed graphs

Petersen proved that every 2k-regular graph can be decomposed into k disjoint 2-factors. I would like to know that is it true that if G is a directed regular graph (d_out(v)=d_in(v)=k), then can G be ...

**1**

vote

**1**answer

103 views

### NP hard problems on UD graphs

I'm reading up on NP hard problems in Unit Disk graphs. I'd like to point out i'm fairly new to this NP hard stuff so i'm trying to get around how to prove something is NP hard.
...

**2**

votes

**1**answer

132 views

### Graph automorphism that swaps two pairs of nodes

Suppose we have two automorphisms on a graph $G$ such that each one swaps a separate pairs of vertices. Is it possible to construct (or prove the existence of) a third automorphism that swaps both ...

**0**

votes

**1**answer

54 views

### Intersection graphs

Does anybody know of a paper which proves that finding the maximum independent set in geometric intersection graphs is NP hard? Even general intersection graphs?

**0**

votes

**0**answers

79 views

### Does the weak Hadwiger conjecture imply the Hadwiger conjecture?

For any cardinal $\kappa$, let $K_\kappa$ denote the complete graph on $\kappa$. We consider the following statements:
(H) If $G$ is a graph and $\chi(G) = \kappa$ then $K_\kappa$ is a minor of $G$.
...

**2**

votes

**1**answer

83 views

### irregular pairs in half graphs - Szemeredi regularity

Szemeredi's regularity lemma is a well-known result about partitioning large graphs into pieces such that most pairs of pieces are "regular". The precise statement takes a bit of detail so I'll just ...

**12**

votes

**0**answers

163 views

### Coloring the edges of a torus graph

Question:Consider the $2k \times 2k$ grid graph on a torus. Is it true that for every $2$-coloring of the edges, there is an antipodal pair of vertices connected by a path that changes colors at most ...

**8**

votes

**1**answer

175 views

### Vertex-primitive graphs with two vertices having almost the same neighbourhood

Hypothesis: Let $\Gamma$ be a vertex-primitive graph with two vertices $u$ and $v$ such that $$|N(u) \cap N(v)|=|N(v)|-1$$
Question: Is it true that $\Gamma$ must either be a complete graph or have ...

**0**

votes

**0**answers

48 views

### Expected number of distinct nodes visited in a directed bipartite graph [migrated]

Let $G = (V,E)$ be a directed bipartite graph with $V = \{I \cup O\}$ where $\left\vert{I}\right\vert = n$ and $\left\vert{O}\right\vert = m$.
All the edges start from a vertex in $I$ and end on a ...

**0**

votes

**1**answer

30 views

### Is it known whether Minimum Cost Multicut is APX-hard?

My questions is concerned with the following problem: Given an undirected graph $G = (V, E)$ and (edge costs) $c \in \mathbb{Z}^E$,
$$\min \left\{ \sum_{e \in E} c_e x_e\ \middle|\ x \in \{0,1\}^E \ ...

**0**

votes

**1**answer

87 views

### How to solve the opposite of max flow problem? [closed]

Each edge in graph has a minimum bound instead of a maximum capacity.
The goal is to find the minimum flow such that every edge in graph has flow larger than it's lower bound.
I tried to use Ford ...

**7**

votes

**3**answers

343 views

### Is there a continuous analogue of Ramanujan graphs?

I think it might help to think of the following definition of a Ramanujan graph - a graph whose non-trivial eigenvalues are such that their magnitude is bounded above by the spectral radius of its ...

**4**

votes

**0**answers

106 views

### minimal polynomial for a graph

I wonder if there is any result relating the degree $d$ of the minimal polynomial of a directed finite graph to any of its topological features - such as its diameter, or any other similar 'natural' ...

**6**

votes

**2**answers

224 views

### Counting valid coordinates

We are given a matrix $D = (d(i,j))_{1 \leq i,j \leq n}$ such that $d(x,z) \leq d(x,y) + d(y,z)$ for each $1 \leq x,y,z \leq n$. It is also known that $d(x,y) \in \mathbb{N}$ (In this question $0 \in ...

**2**

votes

**2**answers

49 views

### Deciding whether a given graph has an f-factor or not!

Given a graph $G$ with $n$ vertices and a function $f$ from $\{1,2,...,n\}$ to non-negative integers, Does there exist an efficient (for example polynomial time) algorithm, that decides whether $G$ ...

**1**

vote

**1**answer

136 views

### Probability of each edge in K-clique [closed]

For $c \in R$ and $k \in N$, $k \geq 3$ let
$p_{k,c} := n^{\frac{−2}{k+1}}log^c(n)$.
I would like to prove that exists $c\in R$ such that every edge in the random graph $G(n,p_{k,c})$ lies in a copy ...

**0**

votes

**1**answer

69 views

### Using upper bound information in graph search

I am using A* (A-Star) to search a graph. A* algorithm takes advantage of the information $h(x)$, which is a lower bound of the distance between a vertex $x$ and the destination vertex.
In other ...

**2**

votes

**2**answers

79 views

### About the roots of the matching polynomial

Can someone kindly give me an expository reference on matching polynomial and its roots? (there is a proof that they are always real?)
I saw these two related discussions,
Roots of matching ...

**2**

votes

**0**answers

60 views

### Are there two-sided $\varepsilon$-expanders with independent sets of size $(1-\varepsilon)n$?

Terry Tao's notes on expander graphs has the following exercise:
Exercise 13 Let $G$ be a $k$-regular graph on $n$ vertices that is a two-sided $\epsilon$-expander for some $n > k \geq 1$ and ...

**1**

vote

**1**answer

204 views

### Probability of connected graph on torus

Let $G = (V, E)$ be a graph on n vertices constructed in the following way:
Each vertex $v \in V$ is positioned uniform randomly in $[0, 1] × [0, 1]$.
Connect two vertices $u, v \in V$ if $d(v,u) ≤ ...

**2**

votes

**0**answers

51 views

### Fixing (non)-independency of a the subfamilies of finitely many events.

I'm would be interesting in any construction of a probability space with n events (n is given), where for every subset of these events, it is also given whether or not, the family is mutually ...

**1**

vote

**0**answers

48 views

### clustering in a graph on boolean functions

Fix some n and k. Consider the following directed graph: vertices are all functions $2^n\rightarrow 2^n$ and a vertex f has an edge to a vertex g for every h such that $f=h\circ g$ and h depends only ...

**2**

votes

**2**answers

203 views

### Find all faces in a graph from list of edges

I have the information from a undirected graph stored in a 2D array. The array stores all of the edges between nodes, e.g. graph[3] might be equal to [1,8,30] and represents the fact that node 3 ...

**3**

votes

**2**answers

256 views

### Graph game minimum vertex degree

Consider the following graph game, given a graph $G=(V,E)$ on $n$ vertices with minimum degree $ >> log(n)$. Players are BR and MA (BR moves first):
BR claims an unclaimed edge from $E$, adds ...

**2**

votes

**1**answer

148 views

### Solving assignment problem using Hungarian method vs min cost max flow problem

The traditional solution for the assignment problem is the Hungarian method - it's complexity is O(V^4) or O(V^3) if using Edmonds method.
However, it can also be reduced to a min cost max flow ...

**5**

votes

**3**answers

103 views

### Minimize distance between centroids of subsets of points

In a n-dimensional space, I want to divide a set of m points into v (non-empty) subsets.
I want to minimize the sum of the pairwise Euclidean distances between the centroids of the resulting subsets.
...

**3**

votes

**1**answer

138 views

### Polygamous stable marriage/ assignment problem

I'm not sure under which 'algorithm' it falls under, but here is the problem:
I need to match each person to 5 people from the opposite gender (each guy gets 5 girls, each girl gets 5 guys). Not all ...