**3**

votes

**0**answers

209 views

### For Ising models on finite graphs, is the gradient of Z (w/r/t coupling and field) easier to compute than Z?

Suppose we have a graph $G$ with $n$ vertices, edgeset $E$, $\mathcal{X}=\{1,-1\}^n$. The partition function of the spin-1/2 Ising model on $G$ is
$$Z(J,h)=\sum_{x\in \mathcal{X}} \exp\left(J ...

**3**

votes

**0**answers

129 views

### Dimension of convex arrangements for hypergraphs

Suppose you have a hypergraph H on n vertices. Let d be the smallest integer such that we can find an arrangement A of convex subsets in Rd so that H represent the intersections of sets in A.
Has ...

**3**

votes

**0**answers

444 views

### Graph recognition software

ISGCI lists a lot of graph classes, many of which are recognizable in polynomial time. Is anyone here aware of actual implementations of these algorithms?

**3**

votes

**0**answers

410 views

### cell complexes and higher graph theory

Suppose that, on an intuitive basis, one defines a "2-graph" $(V,E,F,\partial)$ as a collection of vertices, oriented edges and oriented faces, all of which should be considered as abstract objects ...

**2**

votes

**0**answers

43 views

### What is the densest bipartite graph with unique Hamiltonian cycle?

In a prior post regarding perfect matching, it was stated that the densest graph with a perfect matching cannot have more than $n^2$ edges, if graph has $2n$ vertices.
Analogously, what is the ...

**2**

votes

**0**answers

74 views

### Maximum number of $4$-cycles

Suppose we have a balanced bipartite planar maximum degree $k$ graph.
How many such graphs on $2n$ vertices have at most $f(n)$ maximum number of $4$ cycles for a given function $f:\Bbb ...

**2**

votes

**0**answers

76 views

### Regularity for a bipartite graph

Let $G$ be a bipartite graph with $2^n$ left vertices and $2^n$ right vertices such that:
1) degree of every vertex is not greater then $2^t$
2) number of all edges is greater than $2^{n +t - O(\log ...

**2**

votes

**0**answers

56 views

### When is the graph of cliques isomorphic to the graph itself?

Given a graph $G$ and the set $C_k(G)$ of the $k$-cliques in $G$, one can build a clique graph $H$ whose vertices $c_i\in C_k(G)$ are connected if the vertex sets of $c_i$ and $c_j$ have an ...

**2**

votes

**0**answers

132 views

### Is the class of Heyting algebras originating from directed graphs a variety?

The category RefGph of reflexive directed graphs
is the functor category $\hat{∆}_1=\mbox{Fun}(∆^◦_1,$Set), where $∆_1$ is
the simplex category truncated at level 1.
Hence the poset Sub(X) of ...

**2**

votes

**0**answers

79 views

### Dicks–Dunwoody almost stability theorem

In the book 'Groups acting on graphs' (1989), Dicks and Dunwoody prove the following theorem (paraphrased):
Let $G$ be a group acting on a set $E$, let $E'$ be a subset of $E$ and let $V$ be the set ...

**2**

votes

**0**answers

97 views

### Systematic treatment of folding and valued graphs

I'm going to say beforehand that this question has something of a "am I missing something?" flavor. I'm in that odd position mathematicians often find themselves, where a topic has been addressed ...

**2**

votes

**0**answers

95 views

### Detecting Negative Cycles in Undirected Graphs

I recently faced the problem of quickly detecting negative cycles in undirected, weighted graphs. Resorting to the Bellman-Ford Algorithm, as commonly suggested, turned out to be very inefficient and ...

**2**

votes

**0**answers

111 views

### Formulating shortest path as submodular minimization

I'm curious about the general question of whether any combinatorial optimization problem with polynomial time solution can necessarily be reformulated as minimizing a submodular function.
The answer ...

**2**

votes

**0**answers

121 views

### Which functions preserve the connectivity of graphs/components?

I am somewhat stuck working on an issue and would really love some guidance. I will state the problem, my current state and what led to it in case the solution lies beyond where I was looking
The ...

**2**

votes

**0**answers

58 views

### Blossoms and Colorings

There is a striking analogy between finding maximum matchings in graphs and determining the chromatic number of graphs: both problems are fairly easy for bipartite graphs, but harder, resp. too hard ...

**2**

votes

**0**answers

43 views

### Do product distributions (or graph products) eventually cluster as more products are taken?

Say we have a joint distribution on a finite alphabet $\mathcal{X}\times \mathcal{Y}$. It could be a communication link where we want to send a random message $X$ over a channel, but it gets garbled ...

**2**

votes

**0**answers

57 views

### Planar triangulations for which all distinct 4-colorings consist of exactly 6 Kempe chains

Are there any internally 6-connected planar triangulations other than the icosahedron all of whose distinct 4-colorings consist of exactly 6 Kempe chains, one for each of the 6 color-pairs?
Addendum: ...

**2**

votes

**0**answers

51 views

### The numbers of edge colorings and partitions into vertex covers

Let $G = \langle X \cup Y, E\rangle$ be a $d$-regular bipartite graph such that $|U|=|V|=n$, and assume that $d$ divides $n$. Let $F(G)$ denote the number of proper edge colorings of $G$ using $d$ ...

**2**

votes

**0**answers

282 views

### Maps between graphs defined through laplacian operations

Edit: The views/answers ratio on this question tells me that it was too long. As such, I've stripped out examples and now ask the question in brief. For examples, please ask in the comments or look at ...

**2**

votes

**0**answers

69 views

### Non-existence of commutative rings with many nilpotent elements with some restrictions where matrix powers are efficient

At the moment can't find better reference than "Cycle Enumeration using Nilpotent Adjacency Matrices with Algorithm Runtime Comparisons"
though certainly there are others.
Consider the following ...

**2**

votes

**0**answers

98 views

### Obtaining a quasi-isometry of the 'boundary'

It is well-known that a quasi-isometry induces a homeomorphism on the space of ends of say a locally finite graph for simplicity. Clearly the converse is not true. In other words the concept of ends ...

**2**

votes

**0**answers

55 views

### Modularity in a graph - definition of the random component

This question concerns the definition of modularity in a graph.
Consider a simple, undirected, unweighted graph $G$ with vertices in set $V$ and edges in set $E$ . Between any two vertices there is ...

**2**

votes

**0**answers

74 views

### Even cycle constrained edge coloring

Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every $2t$ simple cycle where $t\in\Big\{1,\dots,2\Big\lfloor\frac{n}2\Big\rfloor\Big\}$ contains atleast $t+1$ ...

**2**

votes

**0**answers

104 views

### From Planar Graphs To Tangent Circles

I have a conjecture:
"For each planar graph with vertices $V_1, V_2,\ldots, V_n$ there exist disjoint circles $w_1,w_2,\ldots,w_n$ in the plane, such that for every $i,j$, $w_i$ is tangent to $w_j$ ...

**2**

votes

**0**answers

51 views

### Maximum cardinality general factor of a graph

Given a graph $G=(V,E)$ and a set of integers $B(v)$ associated to each vertex, a general factor of $G$ is a set of edges $F\subseteq E$ such that the degree of each vertex $v\in V$ in the graph $(V, ...

**2**

votes

**0**answers

42 views

### Is this infinite family of non-trivial snarks resulting from the first Celmins-Swart?

Non-trivial snark is cubic graph with chromatic index $4$, girth
at least $5$ and doesn't to contain three edges whose deletion results in a disconnected graph, each of whose components is nontrivial.
...

**2**

votes

**0**answers

69 views

### Decomposing a weakly chordal graph into disjoint union of co-chordal graphs

A graph G is said to be co-chordal if it is $\bar C_n$-free for any $n \ge 4$. It is weakly chordal if it is $C_n$ and $\bar C_n$ free for all $n\ge 5$. Assume that the induced matching number of $G$ ...

**2**

votes

**0**answers

64 views

### Edge-disjoint path-systems in infinite digraphs

Let $ D=(V,A) $ be a directed graph without backward-infinite paths and let $ \{ s_i \}_{i<\lambda},U \subset V $ where $
\lambda $ is some cardinal. Assume that for all $ u\in U $ there is a ...

**2**

votes

**0**answers

52 views

### Mappings between adaptive networks and Markov processes

Are there any known mappings between adaptive networks models (i.e. graph model representations of networks where the internal vertex dynamics and connectivity topology can change subject to specific ...

**2**

votes

**0**answers

30 views

### Is it true that centrality measures in SNA are indicative for most important vertices only?

I read about the limitations of centrality measures on Wikipedia. It says that centrality measures are good only for identifying top most important nodes in a social network. Their relative values can ...

**2**

votes

**0**answers

63 views

### Is the size of maximum matching in vertex transitive 3-uniform hyper-graph on $n$ vertices always $\Omega(n)$?

What is the best known lower bound on the size of the maximum matching in a vertex transitive $3$-uniform hyper-graph?

**2**

votes

**0**answers

126 views

### About the small set expansion conjecture

Given a graph $G=(V,E)$ and a $\delta > 0$ one wants to calculate $h(G,\delta)=min_{\vert S\vert \leq \delta \vert V \vert } \phi(S)$. ($\phi(S) = \frac{ E(S,\bar{S}) }{d min \{\vert S \vert , n - ...

**2**

votes

**0**answers

89 views

### Linear intersection number and coloring (not chromatic) number

A linear hypergraph is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set and $L\subseteq {\cal P}(X)$ has the following properties:
for $e\in L$ we have $|e|\geq 2$;
if $e_1\neq e_2 \in L$ then ...

**2**

votes

**0**answers

108 views

### the choosing of an independent set in a specific $k$-partite graph

Let $k\geq2$ be an integer, a graph $G=(V,E)$ is called $k$-partite if $V$ admits a partition into $k$ classes such that every edge of $G$ has its ends in different classes: vertices in the same class ...

**2**

votes

**0**answers

40 views

### Isomorphic Hadwiger graphs of connected infinite 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

**0**answers

158 views

### Discrete p-Laplacian

One of the definitions of the discrete (weighted) $p$-Laplacian is the following:
$$\Delta_{p,w}u(x):=\sum_y |u(y)-u(x)|^{p-2}(u(y)-u(x))w(x,y).$$
Consider the one dimensional case. Then the free ...

**2**

votes

**0**answers

100 views

### Isomorphic subcategories of directed graphs and presets

For the purposes of this post, a digraph (directed graph) has neither loops nor multiple parallel edges, and a preset is an ordered pair consisting of a set $S$ and a preorder (viz., a reflexive and ...

**2**

votes

**0**answers

54 views

### Counting labelled graphs according to sets of size 3

In this question we are counting labelled simple graphs. No concept of isomorphism is involved.
Let $G(n,e,t)$ be the number of labelled simple graphs with $n$ vertices, $e$ edges, and $t$ sets of ...

**2**

votes

**0**answers

57 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 ...

**2**

votes

**0**answers

121 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 ...

**2**

votes

**0**answers

75 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 ...

**2**

votes

**0**answers

86 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 ...

**2**

votes

**0**answers

56 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 ...

**2**

votes

**0**answers

63 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

**0**answers

132 views

### Connected graph as connected space

Let $G$ be locally finite graph. By $|G|$, we mean the Freudenthal compactification of the 1-complex $G$, see chapter 8 of "Graph Theory" by Diestel. It is followed from Lemma 8.5.4 of Diestel's book:
...

**2**

votes

**0**answers

134 views

### Minimum number of perfect matchings in a regular bipartite graph

Is there a lower bound on the number of perfect matchings in a $k$-regular bipartite graph?
One can use Hall's marriage theorem and induction on $k$ to derive the lower bound of $k$. I can't come up ...

**2**

votes

**0**answers

52 views

### “drift” of a random graph $G(n,p)$ with $p=\alpha\ln{n}/n$

Suppose $G\sim G(n,p)$ with $p=\alpha\ln{n}/n$ for some large constant $\alpha$. I wish to show a certain "drift" property of $G$, which can roughly be phrased as follows: if $u$ is "not that far" ...

**2**

votes

**0**answers

63 views

### Weighted graph similarity

I have the following problem. Consider an undirected biconnected graph with $n$ vertices and $m$ edges ($n \leq m \leq n(n-1)/2$). The $m$ edges of the graph are then "populated" by integer weights ...

**2**

votes

**0**answers

87 views

### Problems Solvable/Decidable by Counting Shortest Paths in Graphs

This questions is based on a dispute, whether it would be possible to calculate 'nice' routes in Manhattan, if the road network is assumed to be a rectangular grid and, that 'nice' means that there is ...

**2**

votes

**0**answers

88 views

### Disks in Flat Embeddings of Graphs in $\mathbb{R}^3$

Robertson, Seymour and Thomas proved that any linkless graph $G$ has a flat embedding in $\mathbb{R}^3$ (see for example A survey of linkless embeddings). An embedding of $G$ is flat if for any cycle ...