Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, ...

learn more… | top users | synonyms

3
votes
0answers
104 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 ...
3
votes
0answers
123 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 ...
3
votes
0answers
95 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 ...
3
votes
0answers
135 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 ...
3
votes
0answers
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 ...
3
votes
0answers
101 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 ...
3
votes
0answers
98 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 ...
3
votes
0answers
62 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 ...
3
votes
0answers
70 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\ ...
3
votes
0answers
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 ...
3
votes
0answers
65 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?
3
votes
0answers
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]. ...
3
votes
0answers
165 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
0answers
255 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 ...
3
votes
0answers
96 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 ...
3
votes
0answers
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 ...
3
votes
0answers
49 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 ...
3
votes
0answers
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 ...
3
votes
0answers
68 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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
117 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 ...
3
votes
0answers
194 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 ...
3
votes
0answers
199 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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
150 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 $i,j$, ...
3
votes
0answers
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 ...
3
votes
0answers
300 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 = ...
3
votes
0answers
85 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 ...
3
votes
0answers
153 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 ...
3
votes
0answers
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 ...
3
votes
0answers
65 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 ...
3
votes
0answers
305 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 ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
445 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 ...
3
votes
0answers
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 ...
3
votes
0answers
131 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 $N$, ...
3
votes
0answers
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 ...
3
votes
0answers
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 ...
3
votes
0answers
203 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 ...
3
votes
0answers
122 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 ...
3
votes
0answers
253 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 ...
3
votes
0answers
327 views

When is polytope compatible with network flow?

A linear program is the problem of optimizing an linear objective function within some polytope $A$ over $\mathbf R^n$. My question is motivated by the question of when a linear programming problem ...
3
votes
0answers
246 views

Domination number of hamming graphs

The problem of finding the domination number of Hamming graph $H(3, 2n)$ ($n$ is an integer) was given as a homework for my discrete math class. I didn't manage to solve the question. But later the ...
3
votes
0answers
304 views

Computing Quantum Dimensions

Hi, in "Jaeger’s Higman-Sims state model and the B2 spider" by Greg Kuperberg (arxiv:math9601221v1, 1996) there are some quantum dimensions listed in the "Discussion" part. Evidently quantum groups ...
3
votes
0answers
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
0answers
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 ...