# Tagged Questions

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47 views

### Number of Hamiltonian paths in the Hoffman-Singleton graph/Moore graphs?

Does anyone happen to know how many Hamiltonian paths are in the Hoffman-Singleton graph, or have any idea on how I could count such paths?

**3**

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98 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 ...

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75 views

### Graph adjacency grouping with geometric criteria

I start with a list of adjacent tetrahedra, where there are tight seals to one another along faces for two tetrahedra that are adjacent. The vertices belonging to these faces for both tetrahedra are ...

**3**

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80 views

### Is there a Havel-Hakimi for geometric graphs?

Suppose that we are given $n$ points in the plane, with a degree prescribed for each, and the question is whether we can place a geometric graph on them. Is there an efficient algorithm for this?
...

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130 views

### Finiteness for 2-dimensional contractible complexes

While thinking about graph-complex and related operadic stuff, I found a quite interesting (at least for me) question. However, I'm a novice in the algebraic topology, so I'm unable to resolve it by ...

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106 views

### Construction of algebraic curves using line bundles on graphs

In this paper http://arxiv.org/abs/0707.1309 Matthew Baker and Serguei Norine, construct a analogue of the Riemman Roch formula for Lineal Systems defined on graphs. In the paper http://arxiv.org/abs/...

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124 views

### In a random graph which one is more probable, $k$-clique or $k$-core?

Recall that the $k$-core of a graph $G$ is the unique maximal subgraph of $G$ with minimum degree at least $k$.
In an Erdos-Renyi random graph, where the edge selection is independent with ...

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100 views

### Group Travel Salesman Problem

For the following Group Traveling Salesman problem, I'd like to know if there exists some poly-time approximation algorithm with constant approximation factor.
Group TSP is defined as follows: Take a ...

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136 views

### Edit distance vs. canonical adjacency matrix distance

Let $G$ and $G'$ be two simple random graphs on the same set of nodes. Let $d_{edit}$ be the edit distance between $G$ and $G'$. Let $\mathbf{A}$ and $\mathbf{A'}$ be the adjacency matrices of the ...

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104 views

### Synonyms for “labeling” of a graph

In Preprint 1 we write numerical labels 0 or 1 at each vertex of a Dynkin diagram $D$. We call it a labeling of the graph (Dynkin diagram) $D$.
In Preprint 2 we consider an extended (affine) Dynkin ...

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103 views

### The degree/diameter problem for even girth graphs starting with upper bound

I posted this on stackexchange but due to a lack of response there I am posting here.
Let $G$ be a graph with girth $g$, minimal degree $\delta$, maximal degree $\Delta$, and diameter $D$. Define $$...

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99 views

### Does this notion of “$\mathcal{F}$-digraph” appear in the literature?

By a digraph, I mean a quiver with no multiple edges. So in particular:
Loops are okay.
An infinite set of vertexes is okay.
Furthermore, I will tend to identify each digraph with its underlying ...

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63 views

### Small degree vertices in an epsilon-tough graph

We say that a graph is t-tough if by deleting a set if vertices $S$, the resulting graph will have at most $|S|/t$ connected components. We say that a graph is minimally t-tough if the deletion of an ...

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72 views

### The Class of Strong Resolving Graphs of Hamiltonian Outerplanar Graphs

I'll start with a couple important definitions. I'm not sure how well-known any of them are.
Firstly, if $G$ is a graph, and $u, v \in V(G)$, say that $u$ is maximally distant from $v$, denoted $u\ ...

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153 views

### Polynomial-time algorithm solving approximately a generalization of the travelling salesman problem

Take a graph $G$ and a number of sets of nodes of $G$. The problem is to find the shortest path passing through at least one node in each node set. If each node set consists of only one node, the ...

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66 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?

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183 views

### Characterizing graphs with $k$ edge-disjoint minimum diameter spanning trees

Henneberg [1] and Laman [2] characterized graphs which have, after adding any edge, 2 edge-disjoint spanning trees. This was generalized to $k$ edge-disjoint spanning trees by Frank and Szegõ [3]. ...

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166 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 ...

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258 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 ...

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98 views

### What is the significance of the median eigenvalue?

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...

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77 views

### Different definitions of linkless graphs

Robertson, Seymour and Thomas defined linkless embeddings of graphs as follows:
An embedding of $G$ is linkless if every pair of disjoint circuits of $G$ have zero linking number (see here). However ...

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51 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 $K$-...

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98 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 ...

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70 views

### Maximal $k$-chordal subgraph

Recall that a graph is called $k$-chordal if any cycle $C$ of length $> k$ contains a chord, i.e. an edge joining to non-consecutive vertices in $C$. Let $f(n, k)$ be the minimal number of edges s....

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88 views

### Reconstructing a function from its variants that negate one argument

Call two functions $g(x_1,\ldots,x_n)$ and $h(x_1,\ldots,x_n)$ from complex numbers to complex numbers equivalent if they are the same up to the order of their arguments. Formally: there is a ...

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159 views

### Must distinct tree eigenvalues be relatively far apart?

How close to each other can two distinct eigenvalues of a tree be, as a function of the number $n$ of nodes ?
For example, the path $P_n$ exhibits a gap of order $\frac{2\pi^2}{n^2}$ asymptotically ...

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44 views

### Balancing out edge multiplicites in a graph

Let $G$ be a multigraph with maximum edge multiplicity $t$ and minimum edge multiplicity $1$ (so that there is at least one 'ordinary' edge).
Is there some simple graph $H$ such that the $t$-fold ...

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119 views

### When is an induced subgraph of a Johnson graph hamilton-connected?

The Johnson graph $J(n,k)$ has vertices which are the $k$-subsets of $\{1, 2, \dots, n\}$ where two vertices are adjacent iff their intersection has size $k-1$. A graph is hamilton-connected if every ...

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195 views

### vertex transitive and Cayley graphs

(all the graphs alluded to below are finite).
Suppose I gave you a graph, and asked you whether it was vertex-transitive. How hard is that algorithmically?
The second question is: suppose I gave you ...

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201 views

### Partitioning a cubic graph into two induced cycles of equal order

I am aware that deciding the existence of a partition of the vertices of a connected graph $G(V, E)$ into two induced cycles is $NP$-complete(Theorem 2). Induced cycle is a cycle without any chord (...

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186 views

### Hitting edges in graphs at random and let them die with honor

Let $G=(V,E)$ be a finite simple 2-connected graph. Let $B\subseteq E$ be a set of bad edges of size $k:=|B|$. For each edge $e\in E$ we toss a fair coin ($p=1/2$) and if the outcome is head, we hit ...

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79 views

### What is this expander-mixing-type graph property?

Fix $C>0$. I am interested in graphs with the following mixing property:
$$\Big|E(S,T)-\frac{1}{2}|S||T|\Big|\leq C\sqrt{|S||T|\max\{|S|,|T|\}}$$
for every disjoint $S,T\subseteq V$. Note that ...

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152 views

### Concentration inequality for function of independent Bernoulli r.v.'s (related to random graph)

Consider a random undirected graph on a set of $n$ nodes, say $\{1,2,\ldots,n\}$, such that the probability of edge between nodes $i$ and $j$ is $p_{ij}$ (we may assume $p_{ij}=o(1)$ for all $i,j$, i....

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189 views

### Bits required to encode difference between number of subgraphs with odd number of edges and number of subgraphs with even number of edges

Let $H = ( V, E )$ be a $k$-uniform connected hypergraph, with $n = |V|$ vertices and $m = |E|$ hyperedges. Let $O_w$ be the number of edge induced subgraphs of $H$ having $w$ vertices and an odd ...

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304 views

### A problem on graph theory and complex numbers!

Let ${\mathcal G} = ({\mathcal V},{\mathcal E})$ be a simple connected undirected graph with $n$ vertices. Also let $z_1, \ldots, z_n \in {\mathbb C}$ be complex numbers such that
$$
||z_1||=\ldots = |...

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86 views

### Usefulness of polynomial ideals for graphs

Given a graph $G=(V,E)$ on $n$ vertices, one can associate to it the polynomial $f \in k[x_1,\dots,x_n]$ given by $f = \prod_{\{i,j\}\in E}(x_i - x_j)$. Then one can map various graph properties into ...

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154 views

### Maximize weighted vertex sequence subject to neighbour inclusion

Let $G=(V,E)$ be a simple undirected graph on $n$ nodes, with node weights $W = [w_1,w_2,\dots,w_n] \in \mathbb{R}^n$. Define the weight for a sequence of nodes $v_1,v_2,\dots,v_k$ by the average $\...

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241 views

### Beating Kadane's Algorithm

I am seeking some reference on already existing work for the following problem.
Given an $n$-dimensional square matrix $A=DP$ where $D$ is a diagonal and $P$ is a permutation matrix (think of Gaussian ...

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67 views

### Is the $d$-dimensional Arrangement of Trees still $NP$-hard?

The $d$-dimensional Arrangement Problem for general graphs is known to be $NP$-hard since the special case $d=1$ (OLA) already is (Garey et al, [1976]). For Trees however, the one dimensional case can ...

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306 views

### How many trees can be constructed from k vertices using an LCA operator?

Consider the class of rooted trees and suppose I have at my disposal a lowest common ancestor (LCA) operator given by
$$
\textrm{lca}(u,v) = \text{the lowest common ancestor of $u$ and $v$}
$$
for all ...

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80 views

### Is there precedent in the literature for a variant of a random geometric graph where vertices are the centroids of discs placed by random sequential adsorption (RSA)?

Imagine I form a random graph by simulating random sequential adsorption (RSA) of discs (each with the same radius $r$) on $[0,1]^2$ until I cover the plane at a density $\leq U$, where $U \leq (\...

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166 views

### Is there a flow criterion for a graph being planar?

Given the set od propagators (say momenta flows) on a Feynman diagram (flow network), I would like to decide whether this diagram is planar or not.
I know that non-planar diagrams manifest different "...

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448 views

### Minimum weight bipartite graph clique covering

I was wondering if anyone here could give me any pointers as to how to solve the following problem.
Let $B=(L,R,E)$ be a bipartite graph, and $\forall u\in L\cup R$, let $c_u$ be a cost associated to ...

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162 views

### Generalised de Bruijn Graph

I have encountered sets of the following type, consisting of words over a finite aphabet $A$.
If $S$ is such a set, then
$S$ is finite,
No word in $S$ is part of another element of $S$, and
every ...

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132 views

### The mean number of vertices in small connected components of random geometric graphs

I place $N$ points on a circular plane of radius $R$, and draw edges to connect points that are less than or equal to some distance $D$ to form a set of graphs or cliques $G_i$. As a function of $N$, ...

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122 views

### On the faces of the multicommodity flow polytope

Consider the multicommodity flow problem on an undirected graph with k source-destination pairs and specified capacity constraints on the edges. The set of concurrently achievable flows (R_1,R_2,...,...

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189 views

### Bounds on number of small minimal cut-sets in graph?

David Karger developed an algorithm for estimating graph reliability; a key lemma in this algorithm is that if a graph has minimal cut-set size $c$, then the number of cut-sets of size
$\alpha c$ is $\...

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218 views

### Minimum sum of distances of G for spanning tree of G

Let $G=(V,E)$ be a undirected simple graph, and $\Omega(G)$ denote the set of spanning trees of G. Does the next funcion
$$l(G) = \min_{T \in \Omega(G)}\quad \sum_{uv \in E} \mbox{dist}_T(u,v)$$
have ...

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123 views

### Polynomials in graphs

I have part of a physical simulation which I've realised can be modelled using a directed graph where each node is a polynomial. I then calculate this graph by functional composition and summing to ...

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254 views

### Minimal-edge graph with diameter 2 and bounded max degree

Consider all connected simple graphs with diameter $d = 2$ and maximal vertex degree $\Delta$. In my particular practical case $\Delta = 4$, but general problem is much more interesting — probably ...