**10**

votes

**2**answers

440 views

### What is the smallest 4-chromatic graph of girth 5?

It is known that the smallest 4-chromatic graph of girth 4 is the Grötzsch graph (11 vertices). What happens for girth 5?
The Brinkmann graph (21 vertices) has chromatic number 4, girth 5 and is ...

**0**

votes

**0**answers

133 views

### Solving gradient of an especial heat equation

In my research I came up with a gradient of heat equation on a edge-weighted graph as:
\begin{equation*}
\nabla_w T_t(t,w) + T(t,w) . \nabla_w L_w + L_w . \nabla_w T(t,w) = 0
\end{equation*}
where ...

**3**

votes

**0**answers

129 views

### What mathematical models can analyze and optimize systems based on gossip?

I look for a mathematical model that can accommodate, analyze and suggest optimizations for a system that can be humanly described as people gossiping about stuff.
System description:
We have a ...

**3**

votes

**0**answers

180 views

### The Bilu-Linial conjecture and Ramanujan graphs

The Bilu-Linial conjecture claims that every $d-$regular graph has a $2-$lift such that for the signing matrix has its eigenvalues between $[-2\sqrt{d-1},2\sqrt{d-1}]$ (the ``signing matrix" is the ...

**2**

votes

**2**answers

99 views

### Automorphism group of directed complete graph

Given a directed complete graph on $n$ vertices, is there an efficient algorithm for computing its automorphism group? Is there a nontrivial upper bound on the order of its automorphism group? How ...

**0**

votes

**0**answers

82 views

### What is this graph property: number of vertices it takes to see every vertex?

I am wondering what the name is for the following graph property: given a graph $G$ what is the smallest cardinality of $A\subseteq G$ such that every $v\in G$ is connected to some vertex of $A$? I am ...

**0**

votes

**1**answer

154 views

### Are these particular kinds of matrices well known?

Given two positive integers $n$ and $a \leq \frac{n}{2}$ consider a $n \times n$ matrix $A$ such that,
all the diagonal entries are either $a$ or $a+1$
all the non-zero off-diagonal entries are ...

**2**

votes

**1**answer

178 views

### Minimum number of edges to remove to have low degree

I have the following problem (k fixed integer):
Input: Graph G.
Output: Minimum number of edges to remove to G to obtain a graph such that every node has degree at most k.
Do you know the complexity ...

**0**

votes

**0**answers

160 views

### Prove or disprove this upper bound on chromatic number

Let $G$ be a simple connected finite graph and let $v \ge 4$ be the number of vertices, $E$ the number of edges, $\chi(G)$ the chromatic number , $\omega(G)$ the clique number and $\Delta$ the ...

**5**

votes

**0**answers

162 views

### Sets of spreads in graphs

Let $G$ be a graph. A $k$-spread is a set of cliques of order $k$ which partition the vertex set (so $k|n$, where $n$ is the number of vertices).
A partial $k$-resolution of $G$ is a set of pairwise ...

**2**

votes

**2**answers

144 views

### Hamming graph and independent sets

I'm defining the Hamming graph $H(d,q)$ in the usual way, so we have a set $S$ of $q$ elements, the hamming graph $H(d,q)$ has vertex set $S^{d}$ (the set of all ordered $d$-tuples of elements of $S$) ...

**1**

vote

**1**answer

197 views

### Threshold for perfect Matchings in Bipartite graph

Consider the random bipartite graph with vertex classes of size $n$ and each edge being present independently with probability $p(n)$.
I know one way to prove the threshold of a perfect matching is ...

**4**

votes

**1**answer

167 views

### Structure of the stabilizer of a vertex-neighborhood of a vertex-transitive graph

Given a simple, undirected graph and a vertex $v$ of the graph, let $L_v$ denote the set of automorphisms of the graph that fixes the vertex $v$ and each of its neighbors. When the graph is ...

**0**

votes

**0**answers

68 views

### Covering a set in a hypergraph

I'm interested in counting the following. Consider a set $\{v_1,\dots,v_m\}$ of $m$ vertices in the complete $k$-uniform hypergraph on $n$ vertices where $m < k$. I want to know the number of ...

**0**

votes

**0**answers

93 views

### Parallelism degree of a DAG

Let me first give a motivation. Suppose a connected DAG G with one source X and one sink Y. The goal is to find some "bottleneck" node between X and Y, i.e. node through which every path from X to Y ...

**1**

vote

**1**answer

80 views

### Spectral radius of a time-varying matrix with strictly positive increment of the matrix's entry

Consider a time varying non-negative matrix $A(t)$ and its spectral radius $\rho(A(t))$ being the largest eigenvalue of $A(t)$ and $t$ denotes the time. If $A(t)$ changes over time with each time a ...

**18**

votes

**5**answers

1k views

### Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?

**11**

votes

**3**answers

932 views

### What is a “Ramanujan Graph”?

I noticed an apparent conflict in the definition in literature about what is a "Ramanujan graph, which I was wondering if someone could kindly clarify.
(1)
The Hoory-Linial-Wigderson review on ...

**10**

votes

**1**answer

255 views

### Are there non-trivial graphs that uniquely embed to hypercubes?

The $n$-dimensional hypercube is the graph formed by $0$-$1$ sequences of length $n$ where two vertices are adjacent if they differ at only one place.
The weight of a sequence is the number of $1$'s ...

**2**

votes

**2**answers

202 views

### Removing subtrees

Let $T$ be a complete infinite rooted binary tree. Is it possible to remove (infinitely many) subtrees of $T$ and get a subgraph $G$ such that:
$G$ has no complete subtrees (the graph below any ...

**4**

votes

**1**answer

224 views

### Prime labelling of graphs

A prime labeling of a graph is an injective function $f: V(G) \to \{1, 2, ..., |V(G)|\}$
such that for every pair of adjacent vertices $u$ and $v$, $\text{gcd}(f(u), f(v)) = 1$ (labels of any two ...

**2**

votes

**1**answer

120 views

### How to construct a graph with arbitrarily large girth and large chromatic number? [closed]

Erdos theorem says it is possible and it is not so easy. What is the general procedure to construct graphs like Grötzsch graph?

**0**

votes

**0**answers

20 views

### Looking to derive bound for modulus of harmonic eigenfunction on weighted graph

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus:
$\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$
and wish to lower ...

**5**

votes

**1**answer

200 views

### Algorithm to minimally connect line segments in Euclidean plane

Suppose you have finitely many line segments in the Euclidean plane. How do you "connect them to form one chain of line segments of minimal length?"
More formally and generally, what I'm looking for ...

**2**

votes

**0**answers

48 views

### finding dominating cycles in $2K_2$-free graphs

A cycle $C$ in a connected graph $G$ is called dominating if its complement $V(G)-V(C)$ is an independent set. H.J. Veldman proved in 1983 (Disc. Math. v.43, 281-96) a general result that in ...

**4**

votes

**1**answer

263 views

### Endomorphisms and almost all graphs

Is it known what fraction (almost all?) of graphs have a trivial endomorphism monoid? I can't seem to find any reference to the question. Maybe it's related to the question: what fraction of graphs ...

**1**

vote

**0**answers

60 views

### Is there an elliptic Harnack equality for directed graphs?

The elliptic Harnack inequality for undirected graphs was proven by Delmotte in the paper "Inegalite de Harnack elliptique sur les graphes" (French, ...

**4**

votes

**1**answer

111 views

### Determining the number of hamiltonian paths of $K_n-C_n$

I would like to know information regarding the function $h(n)$ where $h(n)$ is the number of hamiltonian cycles the graph $K_n$ has after removing the edges that make up a hamiltonian cycle of $K_n$. ...

**2**

votes

**1**answer

181 views

### Edge density of triangle-free graphs

Let $G$ be a finite, simple, loopless graph with $|V(G)|=n$. We define its edge density as $$ed(G) := \frac{|E(G)|}{n \choose 2}.$$
Moreover we set $$d_n := \text{max}\big\{ed(G): G \text{ is a ...

**1**

vote

**0**answers

44 views

### Rate of convergence of graph-theoretic quantity to fractional graph-theoretic counterpart

Let $G^n$ denote the OR product of a graph with itself $n$ times, i.e. the graph which has an edge between distinct vertices $(v_1,v_2,\ldots,v_n)$ and $(u_1,u_2,\ldots,u_n)$ if there exists some $i$ ...

**9**

votes

**1**answer

187 views

### Universal graph homomorphisms

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such ...

**17**

votes

**1**answer

309 views

### Lens spaces and generalized Petersen graphs

Recently I came across this mathoverflow question, in which the number of homeomorphism classes of 3-dimensional lens spaces $L(p, q)$ is computed as a function of $p$. Using the OEIS, I found a ...

**10**

votes

**1**answer

443 views

### Cauchy-Schwarz proof of Sidorenko for 3-edge path (Blakley-Roy inequality)

Is there a "Cauchy-Schwarz proof" of the following inequality?
Theorem. Given $f \colon [0,1]^2 \to [0,1]$, one has
$$
\int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) ...

**1**

vote

**2**answers

184 views

### Proving a random bipartite graph contains a perfect matching

I have the following problem
consider a random bipartite with vertex classes $A$ and $B$ of size $|A|=|B|=\mathrm{log}^{2}(n)$ graph in which every possible edge is chosen independently with ...

**31**

votes

**9**answers

3k views

### Generalizations of the Four-Color theorem

The four color theorem asserts that every planar graph can be properly colored by four colors.
The purpose of this question is to collect generalizations and strengthenings of the four color theorem ...

**6**

votes

**1**answer

267 views

### Do perfect matching(s) have signatures in the graph eigenvalues?

If the edges of a bipartite graph are such that they can be seen as a disjoint union of perfect matchings then will this somehow reflect in the eigenvalues of the Laplacian?
It would be helpful to ...

**0**

votes

**0**answers

62 views

### Computation on Random Bipartite graphs

I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...

**3**

votes

**1**answer

62 views

### Minor ordering for finite graphs

Let $\mathcal{G}_{<\omega}$ be the set of graphs $G = (V,E)$ such that $V = \{0,\ldots,n\}$ for some $n \geq 0$ and $E \subseteq \mathcal{P}_2(V) = \{\{a,b\} : a,b \in V \textrm{ and } a\neq b\}$. ...

**1**

vote

**0**answers

67 views

### Matchings in random bipartite graphs

I was wondering if anyone could point me in the direction of a text or paper which would help deal with the following problem
Suppose i am given a $K_{\mathrm{log}(n)} \times K_{\mathrm{log}(n)}$ ...

**4**

votes

**2**answers

164 views

### Graphs whose degree vectors coincide for all powers of their adjacency matrices

Let symmetric $A,B \in \{0, 1\}^{n \times n}$ denote the adjacency matrices of two simple graphs. Further let $\mathbf{1}$ denote the all-one-vector.
Now assume that $A^k \mathbf{1} = B^k \mathbf{1}$ ...

**0**

votes

**1**answer

84 views

### What is the standard name of an edge-graph

Given a graph $G=(E,V)$, I construct a graph $G'$ where the vertices of $G'$ are given by the edges of $G$ and say that two edges of $G$ are neighbors in $G'$ if they have a common vertex.
Is there a ...

**2**

votes

**2**answers

165 views

### “Homomorphism fingerprint” for graphs

Let $G, H$ be simple, undirected graphs without loops. We say that $G, H$ have the same homomorphism fingerprint if $|\text{Hom}(X, G)| = |\text{Hom}(X, H)|$ for all graphs $X$. (By graph ...

**3**

votes

**0**answers

110 views

### Cores of infinite graphs

Let $\kappa$ be a cardinal and let $\textrm{Grph}(\kappa)$ be the set of graphs $G = (V,E)$ such that $V \subseteq \kappa$ and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$.
We ...

**4**

votes

**1**answer

166 views

### What can be said about graphs if there are homomorphisms in both directions?

Let $G,H$ be simple undirected graphs without loops such that there are graph homomorphisms $f_1:G\to H$ and $f_2: H\to G$.
An easy argument shows that we have $\chi(G) = \chi(H)$, even for graphs ...

**1**

vote

**0**answers

125 views

### NP hard problems on geometric graphs

I have posted this question before but i don't feel i expressed my confusion clearly enough. So i would like to try and explain again. This is a proof of the minimum vertex cover for unit disk graphs ...

**2**

votes

**0**answers

69 views

### Total number of spanning trees of a set of graphs

Given an undirected graph G with $n$ nodes, we can compute its number of spanning trees in polynomial time using Kirchhoff's matrix-tree theorem. Now consider a more complicated setting, in which each ...

**9**

votes

**4**answers

347 views

### Are these three different notions of a graph Laplacian?

I seem to see three different things that are being called the Laplacian of a graph,
One is the matrix $L_1 = D - A$ where $D$ is a diagonal matrix consisting of degrees of all the vertices and $A$ ...

**1**

vote

**0**answers

64 views

### Max flow with minimal requirements algo problem

While applying the algorithm to solve the max flow of the network with minimal requirements on edges, I have encountered a problem.
The algorithm states:
For graph G
create an edge from target to ...

**1**

vote

**1**answer

100 views

### n-cube connectivity problem

Given $n\geq 2$, let us consider the n-cube $H_n=(V,E)$, i.e. vertex set $V$ is $\{0,1\}^n$.
Here, the edges of $H_n$ are directed, oriented by set inclusion, i.e., $(x,y)\in E$ iff $x\subseteq y$ and ...

**7**

votes

**2**answers

124 views

### Realizing a monoid as $\mathrm{End}(G)$ for some graph $G$

This question relates to Realizing groups as automorphism groups of graphs.
Given a monoid $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?