# Tagged Questions

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76 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**

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35 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 ...

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57 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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46 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 ...

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61 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 ...

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54 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 ...

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64 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 ...

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53 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 ...

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73 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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127 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:
...

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75 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 ...

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32 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" ...

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48 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 ...

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68 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 ...

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85 views

### second smallest eigenvalue Laplacian - submodular set function

Let $G$ be a connected unweighted undirected graph. In addition, let $\lambda_2(L(G))$ be the second smallest eigenvalue of the Laplacian matrix of graph $G$. Is $\lambda_2(L(G))$ a submodular set ...

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65 views

### When polynomial GI implies polynomial (edge) colored GI?

(edge) colored graph isomorphism is GI which
preserves the colors (of edges if it is edge colored).
There are several reductions using transformations/gadgets
of (edge) colored GI to GI. For edge ...

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49 views

### Looking for similar centrality measurement on graph

I'm working on a graph problem somehow related to centrality measurement. Given an undirected, unweighted tree $T$ and a vertex $v$, let $D_i(v)$ be the set of vertices in $T$ that are i hops from ...

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72 views

### Properties of a smallest tournament with domination number $k$

For some tournament $T$, let $\gamma(T)$ denote the cardinality of a smallest dominating set of $T$.
Denote by $f(k)$ the minimum number of vertices of a tournament $T$ having $\gamma(T) = k$.
From ...

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188 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 ...

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49 views

### Regular graphs with unimodal subdegrees that are not distance-regular

Distance regular graphs are known to exhibit the following property: starting from an arbitrary vertex $\alpha$, let $k_i$ denote the number of vertices at distance $i$ from $\alpha$ (in terms of ...

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66 views

### Characterize the equivalence class of bipartite graphs obtained from each other by elementary row operations on their adjacency matrices

Let $M$ be an $m\times n$ matrix real matrix. Let $G$ be a bipartite graph, with partitions $A$ and $B$, such that $|A|=m$ and $|B|=n$. A node $i\in A$ is linked to a node $j\in B$ if and only if ...

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59 views

### Smallest distribution of points with genuinely different clusterings

An hierarchical clustering algorithm for (finite) sets of points in a given metric space is essentially determined by its linkage criterion, which defines the distance between arbitrary (finite) sets ...

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92 views

### Proofs about Forbidden Minors

I am working on a problem which involves proving that a particular graph is a forbidden minor of the class of graphs that i am working on.
Now i read kuratowskis theorem for planarity but i still ...

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109 views

### Is there an universal (dis)similarity measure between two structures?

I'm always wondering is there an universal (dis)similarity measure
between two structures (let's say between two undirected simple
graphs)? I mean, not "the measure with universal parameter that we
...

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53 views

### Minimal set of 2-2 Pachner move null sequences on a (nonplanar) trivalent graph?

A "null sequence" is of course a sequence of Pachner moves (inside a closed
area) that doesn't change the trivalent graph. E.g. doing the same Pachner move
twice (leads to orthogonality of 6j symbols) ...

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84 views

### Counting regular Hypergraphs

The problem of counting regular graphs on $n$ vertices is notoriously hard. It seems like counting regular hypergraphs on $n$ vertices should be much easier (I am placing no uniformity condition). ...

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101 views

### Hamiltonian Matroids

Similar to graphs, a Matroid $M$ is said to be Hamiltonian if there is a base $B$ of $M$ and $e \in M-B$ such that $B + e$ is a cycle of $M$. Is there any literature on this?
EDIT: Actually my ...

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140 views

### If two graphs have same Laplacian spectrums, are they the similar graphs?

If graph A and graph B have exactly the same Laplacian spectrums, can I just say they are same graphs? (only with scaling edge weights and node index reordering) I don't know if there is any explicit ...

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137 views

### The Turán problem for graphs with given chromatic number

The ordinary Turán problem for graphs asks, "Given a graph $H$, if $G$ is an $H$-free graph on $n$ vertices, what is the largest number of edges that $G$ can have?" As is well known, if $\chi(H) = r ...

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221 views

### On the existence of Graph Monomorphism

A graph monomorphism is an injective graph homomorphism. Determining existence of Graph monomorphism between graph pairs is computationally hard.
Assume we talk only about classes of undirected ...

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112 views

### Bipartite independence number

Consider a balanced bipartite graph $G=(U,V,E)$, i.e., a bipartite graph with $|U|=|V|$. An independent set $I$ of $G$ is balanced if $|I \cap U| = |I \cap V|$.
The bipartite independence number of ...

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90 views

### Groups acting on non-locally-finite trees with independence and specified local actions

Suppose I have a biregular tree $T_{m, n}$ (not necessarily locally finite), with distinct cardinal numbers $m, n$, so Aut$(T_{m, n})$ acts on $T_{m, n}$ without inversion. Let $V_m$ be those vertices ...

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287 views

### Why the number of the vertices is even?

Let $G$ be a graph with $n$ vertices and $V$ the vertex set. Suppose that for each non-empty subset $W$ of $V$ there exists an element $\omega \in V$ ( maybe in $W$ or not) such that the degree of ...

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93 views

### A surjective graph homomorphism between 4-valent graphs

Let $K$ and $L$ be $4$-valent planar graphs with Euler circuits (where the choice of the next edge in the circuit consists of the edge which does not share a face with the edge before), with $|V(K)| = ...

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92 views

### Reachability in dynamic random graphs

There has been some advice on how to ask questions. So I reask my question.I have a problem related to graph theory, random graphs or random geometric graphs in special that really confuses me. ...

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71 views

### Effect of removing a Hamiltonian cycle on the Laplacian spectrum

Notation: $\lambda_{\max}(G)$ is the largest eigenvalue of the Laplacian matrix of the graph $G$ (aka the Laplacian index of $G$).
Now suppose $G$ is a Hamiltonian graph with Hamiltonian cycle $C$.
...

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55 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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115 views

### Optimization over Spectral Laplacian in cycles and trees

Is there any idea on how one can deal with an optimization problem of sum of k largest eigenvalues(min) of Laplacian matrix of a simple cycle or tree?
I would like to use semidefinite programming for ...

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470 views

### Is there a name for this graph?

I'm trying to find out whether the following graph has a name: Let $W$ be an $n$-dimensional vector space over $GF(q)$. The vertices of the graph are all the subspaces of $W$. Two subspaces $W_1$ and ...

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251 views

### Incremental minimum spanning tree

Given a connected graph $G=(V,E)$ with a weight function $w:E\to\mathbb{R}$ and a subset $E_0\subseteq E$ such that the subgraph $(V,E_0)$ is connected, I am looking for a sequence $E_0\subseteq ...

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88 views

### Shortest loop containing 0 in continuum percolation

I am interersted in continuum percolation with intensity $\lambda>0$. Formally, let $X$ be a Poisson point process in $\mathbb{R}^d$ with intensity $\lambda$ and $G$ the graph obtained by ...

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137 views

### Clique number of a $k$th power of a graph in terms of maximum degree?

Let $G$ be a graph of maximum degree $d$. Are there any known/easy bounds on the clique number of $G^k$, in terms of $d$ and $k$? I would like something at least epsilon better than the bound on the ...

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292 views

### Partitioning the vertex set of a graph with a large independent set

Let $G$ be a graph on an even number of vertices, say $2M$. Assume that the largest independent set in $G$ has at most $M$ elements. Is it true that there exists a set of $2m$ vertices (for some ...

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71 views

### Minimum relay nodes in graph

I have the following problem. I would like to know if this reduces to some standard problem in Graph theory. Any suggestions are much appreciated.
I have a multi-cast network with 1 source (denoted ...

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139 views

### Geodesics in polyhedral graphs

Let $e = \lbrace u,v\rbrace$, $e' = \lbrace v,u'\rbrace$ be edges of an undirected graph $G$ and $ee'$ be the path from $u$ through $v$ to $u'$. The following defintions make sense for every graph ...

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60 views

### Presentation of tree decompositions (and related concepts) in terms of continuous maps?

A tree decomposition of a graph $G$ is commonly defined in terms of a tree $T$ with the following structure:
Each vertex $t \in V(T)$ is associated to a set $X_t \subseteq V(G)$;
The union ...

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69 views

### Maximizing the minimum outdegree of digraph without $m$ cycle

Let $G$ be a simple digraph on $n$ vertices without a directed cycle of length $m$
(it may have directed cycles of length less than $m$. The cycles need not be simple).
How large the minimum ...

**2**

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151 views

### Structure of almost all bipartite graphs

I am studying some properties related to bipartite graphs and it would be useful for me to know if there is anything known about the structure of almost all bipartite graphs. For example, is it true ...

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160 views

### A primal-dual (double) circle packing (coin graph) question

I know that any 3-connected simple planar graph with a designated outside face (outer face) has a primal-dual (double) circle packing (Brightwell-Scheinerman Theorem).
Q1- But I am not sure whether ...

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180 views

### A natural problem on “cartesian union” of set families (hypergraphs). Does anybody know NP-complete problems related to the notion of cartesian product?

I'm curious whether the problem below is NP-complete.
I provide two simple definitions and one example at first.
Definition 1.
Let $\langle {\cal{S}}_i\rangle\substack{i\in I}$ and $\langle ...