**0**

votes

**0**answers

57 views

### How many edges are in the kth-iterated common neighborhood knight's graph?

For terminology see https://www.academia.edu/2180382/The_Common_Neighborhood_Graph_and_Its_Energy)
It has been known for quite some time by players that a vertex whose corresponding square is in the ...

**6**

votes

**0**answers

69 views

### Set system with prescribed intersection sizes

Questions: What is the asymptotic maximal size of a $4$-uniform (every set has 4 elements) set system $\mathcal{A}$ of subsets of $[n]$ such that, no two sets have size of their intersection $2$?
In ...

**1**

vote

**1**answer

96 views

### edge graph reconstruction conjecture : set vs multi set

Why is the edge reconstruction conjecture stated with the deck defined as the multi set of graphs formed by deleting one edge? Can someone give an example of two graphs such that the edge deleted ...

**7**

votes

**2**answers

264 views

### What is the number of noncrossing acyclic digraphs?

A noncrossing graph on $n$ vertices is a graph drawn on $n$ points numbered from $1$ to $n$ in counter-clockwise order on a circle such that the edges lie entirely within the circle and do not cross ...

**5**

votes

**1**answer

129 views

### Graphs where each edge belongs to the same number of 1-factors

Let $G$ be a simple connected graph that has at least one 1-factor. We'll define:
$G$ has property A iff it is edge-transitive.
$G$ has property B iff each edge belongs to the same number of ...

**2**

votes

**0**answers

162 views

### Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate
polynomial which counts the number of colorings of $G$
with $x$ colors for natural $x$.
Graph is not $k$ colorable iff $P(G,k)=0$.
The ...

**1**

vote

**1**answer

120 views

### Vertex transitive and edge transitive and line graph

How can we find the proof of the following statement:
An undirected graph is edge transitive if and only if its line graph is vertex transitive.

**25**

votes

**3**answers

588 views

### Removal of non-isomorphic edges results in the same graph

There exists a (simple unlabeled) graph on 6 nodes with a pair of non-isomorphic edges (i.e., there is no graph automorphism that sends one edge into the other) such that removal of either of them ...

**10**

votes

**2**answers

482 views

### Is every knot unavoidable in the embeddings of some graph?

Is it the case that, for any given knot $K$,
there exists some graph $G$ whose every embedding into $\mathbb{R}^3$
(or into $\mathbb{S}^3$)
contains a cycle that realizes $K$?
I know the ...

**3**

votes

**1**answer

211 views

### How hard is a variant of graph automorphism problem?

I'm interested in a variant of graph automorphism problem (which is prime candidate for $NP$-Intermediate problem).
Restricted GA
Input: Given an undirected graph $G(E, V)$, and $\epsilon |V|/2$ ...

**0**

votes

**0**answers

41 views

### Complexity of graph isomorphism in $(P_4 \cup K_1,\overline{3K_2})$-free graphs

Related to this question where isomorphism preserving
transformation maps triangle-free graphs to $(P_4 \cup K_1,\overline{3K_2})$-free graphs.
What is the complexity of graph isomorphism in $(P_4 ...

**3**

votes

**1**answer

104 views

### Graph transformation related to graph isomorphism

Basically got graph transformation related to graph
isomorphism.
Define $G \to G'$. $V(G')=V(G) \cup E(G)=\{v_1\ldots v_n\} \cup \{e_1\ldots e_m\}$. Call $v_i$ vertices $v'$ and $e_i$ vertices $e'$.
...

**4**

votes

**1**answer

492 views

### Surprising connection between linear algebra and graph theory

What is the intuition for linear algebra being such an effective tool to resolve questions regarding graphs?
For example, one can determine if a given graph is connected by computing its Laplacian ...

**0**

votes

**0**answers

24 views

### Splitting lemma for digraph and preserving local rooted-edge connectivity?

Let $G$ be a directed graph. $\lambda(x,y,G)$ is the maximum number of edge disjoint paths from $x$ to $y$ in $G$
The local $r$-rooted connectivity of $x$ in $G$ is $\lambda(r,x,G)$.
The global ...

**2**

votes

**1**answer

125 views

### Minimum length path touching $n$ circles

Given $n$ non-overlapping circles of radius $1$ (i.e., the distance between the centers of any two circles is greater than $2$), how to find the minimum length path (the path can be of any form) that ...

**3**

votes

**0**answers

40 views

### Algorithm to construct metric space endomorphism with controlled square

Given a finite metric space $(M,d)$ with parameters $K \geq 1$ and $\epsilon > 0$, I'd like to algorithmically check for the existence of a non-identity map $\phi:M \to M$ which happens to be ...

**0**

votes

**1**answer

72 views

### N random walkers that hit node v in a graph

Consider a finite, undirected graph G, with uniform edge weights. Assume that there are n number of random walkers that will start at different nodes (lets say n=3, hence the random walkers will start ...

**2**

votes

**1**answer

83 views

### About the diameter of a graph after removing orientation

This question was posted a few days ago on the Mathematics StackExchange, but so far it has not been answered. Let $G$ be a strongly connected directed graph of diameter $D$, and suppose that we ...

**4**

votes

**0**answers

35 views

### What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

The Tutte polynomial
is a bivariate polynomial with positive integer coefficient which is a graph
invariant and can be defined recursively.
Evaluating it is $\#P$-complete even when restricted to ...

**1**

vote

**2**answers

173 views

### Strongly connected DAG from any connected undirected graph?

I have the following question. It seems likely to be true - can anyone provide a standard reference?
Given:
A connected, undirected graph.
Question 1:
Can we assume a single direction for each edge ...

**7**

votes

**0**answers

107 views

### Algorithms for computing the Resilience of Graphs

The definition of resilience with a graph $G$ w.r.t to a monotone property $\mathcal{P}$ is well known.
(Global resilience) Let $\mathcal{P}$ be an increasing monotone property. The global ...

**3**

votes

**0**answers

84 views

### Node covering in a random graph

Given $N$ nodes randomly placed in a $D\times D$ area, i.e., the position of each node is randomly chosen. Assume that both $N$ and $D$ are sufficiantly large.
An agent can move in the area at ...

**0**

votes

**0**answers

97 views

### Reduction from permanent to $(0,1)$-permanent and implication of $P \ne NP$

Valiant
shows reduction from counting the solutions of CNF formula $F$,$\#SAT(F)$
to computing permanent where $ Perm(A)= 4^{t(F)}\cdot \#SAT(F)$
for certain efficiently computable $t(F)$ and matrix ...

**6**

votes

**2**answers

173 views

### Find multiple non-adjacent paths in a graph

Consider a non-directed graph. I want to find as many non-adjacent paths as possible from a source $s$ to a destination $t$. Two paths $P_1$ and $P_2$ are said to be non-adjacent to each other if none ...

**4**

votes

**1**answer

114 views

### Product of geodesic distances

I'm working on trying to show this, but can't seem to get started. No guarantees that it is true, but other conditions on the adjacency matrix that make it true or a counter example are helpful. ...

**0**

votes

**2**answers

80 views

### different way of selecting a random graph

Consider having a 'base' graph $G=(V,E)$ and selecting each vertex with independent probability $p$ and having the induced subgraph of $G$ with all 'selected' points as your random graph. Has this ...

**0**

votes

**4**answers

111 views

### about the structure of components of tensor product if more than one bipartite graph is taken

I was reading about tensor product of graphs. We know that if we take tensor product of n graphs and want this product to be a connected graph then at most one graph should be bipartite. In the book ...

**3**

votes

**1**answer

105 views

### Graphs of lines on del Pezzo surfaces

Let $k$ be an algebraically closed field. To any del Pezzo surface $S$ over $k$ we may associate its graph of lines, which has one vertex for each line and an edge (with multiplicity if required) ...

**2**

votes

**1**answer

154 views

### Clique problem for regular graphs

I am looking for NP complete results for cliques in regular graphs. For example is the general problem of determining if a regular graph on n vertices has an n/2 clique NP-complete? (obviously the ...

**0**

votes

**0**answers

42 views

### Paths on Cartesian products of graphs satisfying linear constraints

Assume integers $d > r > 0$ and a connected graph $G$ with $d$ vertices. Every point on the $r$-fold Cartesian product of $G$ with itself, $G^{\square r}$, is equivalent to a dimension-$d$ ...

**1**

vote

**0**answers

72 views

### Empty node in cactus construction

Is there a necessary condition for not having empty node in the construction of the cactus of the minimum cuts of a graph?
In particular is there a necessary condition for not having empty k-junction ...

**0**

votes

**0**answers

44 views

### Correlation between attributes in a binary graph

Given an unrooted binary tree whose leaves are vertices of degree one that are labelled bijectively by a set $S$. We define a categorical attribute $A$ ($|A|<<|S|$) and each leaf is assigned a ...

**1**

vote

**0**answers

105 views

### Find a path that covers as many nodes as possible

I have the following interesting problem. Given a graph $G$, an agent starts to mark nodes in $G$ in the following way: it marks all nodes within distance $d$ from it. Now the question is to find the ...

**4**

votes

**1**answer

109 views

### Expected number of connected components in a random graph

For a random graph G(n,p) what is the expected number of connected components? What is the probability distribution of this value?
I'm specially interested in what happens for small values of p, ...

**-4**

votes

**1**answer

148 views

### Bipartite graph [closed]

First of all, thank you for your time to reading my post.
I am a researcher but not a mathematician, i have some difficulties in solving a math problem, that why i am here to ask your help. I just ...

**0**

votes

**1**answer

154 views

### Kneser graphs eigenvalues

Basically, I want to prove that, in the Kneser graph (wikipedia has a good definition),$K_{n, m}$, if $n_{-}(A(G)) $ and $n_{+}(A(G))$ denote the number of negative and positive eigenvalues of A(G) ...

**6**

votes

**2**answers

198 views

### Conjecture: for perfect graphs the fractional chromatic index rounded up equals the chromatic index

Let $\chi'_f(G)$ be the fractional chromatic index.
Based on limited experiments (up to 8 vertices and few larger graphs),
I suspect:
Conjecture For perfect graphs $\lceil \chi'_f(G) \rceil = ...

**2**

votes

**1**answer

92 views

### Estimate for the travelling salesman problem for balls inside a grid

This question is probably easy but I only have "tedious case checking" proof strategy in sight, and I'm sure there should be a reference lying around...
The question concerns the TSP problem (with ...

**1**

vote

**0**answers

94 views

### random walk with reflecting barriers [closed]

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are (same ...

**5**

votes

**2**answers

202 views

### Roots of matching polynomial of graph

At the end of this preprint, I make the following conjecture concerning the roots of the matching polynomial:
If a graph $G$ is connected and contains a cycle, then the spectral radius of $G$ ...

**0**

votes

**0**answers

82 views

### Tactics/strategies to win the chip firing game between two-players

Given a connected, undirected graph, two players (black and red) play the following game:
Note: Initially, all vertices do not contain any tokens.
To begin the game, each player (black or red) ...

**0**

votes

**1**answer

79 views

### Finding node-disjoint routes with mutually exclusive nodes in graphs

I have the following problem. I would like to know if it reduces to some standard problem in Graph theory. Particularly, I would like to know whether it is NP-hard, if yes, how to prove its ...

**1**

vote

**0**answers

46 views

### Simulation of disassortative random graphs

Recently I have been trying to find a succinct algorithm for generation of disassortative networks. The best I have found is the algorithm by Newman described in his paper "Mixing patterns in ...

**1**

vote

**0**answers

51 views

### Polynomial problems in graph classes defined by forbidden induced cyclic subgraphs

Let $C$ be a graph class defined by a finite
number of forbidden induced subgraphs, all
of which are cyclic (contain at least one cycle).
Are there graph problems that can be solved in
...

**3**

votes

**3**answers

300 views

### Turan's theorem for connected graphs?

Using a small modification to Turáns theorem we can find the minimum amount of edges a graph $G$ on $n$ vertices must have so it does not have an independent set of size $k$. Is there a similar result ...

**1**

vote

**0**answers

52 views

### Asymptotic results in unbalanced left $d$-regular expander graphs

Let $U = [n]$ and $V = [m]$ be sets of nodes with $n > m$ and $E = U\times V$ be a set of edges. Let $\mathcal{N}(S)$ be the set of neighbors of a subset $S$ from $U$ or $V$.
Call a graph $G = (U, ...

**6**

votes

**0**answers

132 views

### The Universal Labeling of graph

The universal labeling of a graph $G$ is a labeling of the edge set in $G$ such that in every orientation $\ell$ of $G$ for every two adjacent vertices $v$ and $u$, the sum of incoming edges of $v$ ...

**8**

votes

**1**answer

166 views

### Spectral lower bounds on the diameter of a graph

There is such a bound, due to Mohar and McKay, using the second-smallest eigenvalue of the Laplacian $\lambda_{2}$:
$$Diam \geq \lceil \frac{4}{n\lambda_{2}} \rceil.$$
This bound is very elegant but ...

**1**

vote

**1**answer

98 views

### How to model a time-discrete heat equation on a graph?

I would like to know how one can set up a time-discrete model for the heat equation on a graph?
Is this problem using the heat kernel equation on a graph?

**0**

votes

**0**answers

70 views

### Heat equation with graph laplacian

I would like to start with considering the time-dependent heat equation on a connected graph and consider its Laplacian matrix.
Suppose we have a connected graph with unknown temperature on vertices. ...