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
34 views

eigenvalues of cycle and its complement [on hold]

I am trying to find the eigenvalue of cycle graph and its complement. How to simplify.Suppose $\omega^{1}+\omega^{n-1}=2cos (2\pi/n) $, then, $\omega^{\frac{n-1}{2}}+\omega^{\frac{n+1}{2}}=?$ Is it ...
1
vote
1answer
47 views

approximate diameter of polytopes in high dimensions

I just came across the following problem: Let us consider the unit corner of the n-cube $$ \Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 ...
0
votes
1answer
78 views

Are constructive characterisations of k-regular (simple) graphs known?

By a constructive characterisation I mean a theorem giving a list of base graphs and a list of operations such that every graph in a given class is generated from the base graphs by applying some ...
8
votes
0answers
61 views

What are some useful invariants for distinguishing between random graph models?

Quite a few probabilistic algorithms for generating random graphs exist in the literature, such as: The Erdos-Renyi model The Stochastic Block model The Watts-Strogatz model The Barabasi-Alber model ...
2
votes
0answers
21 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 ...
4
votes
1answer
53 views

Maximum and minimum diameter of categorical graph product

Let $G_i$ be connected finite simple undirected graphs with diameter $d_i$ for $i=1,2$. Assume that $G_1\times G_2$ is connected. (Here $G_1\times G_2$ denotes the categorical product.) In terms of ...
-1
votes
0answers
29 views

Any polynomial-time algorithm for hypergraph bisection? [on hold]

I work with hypergraph partitioning. I want to divide a complete weighted hypergraph into 2 parts using cut-net metric, a sum of all edges cut, and connectivity metric. Is there a polynomial-time ...
1
vote
0answers
69 views

What characteristic of a graph depend on the vertex labeling?

Different labeling on a graph produces class of isomorphic graphs. Two isomorphic graphs possess similar characteristic such as connectivity, degree distribution of vertices, equality of spectrum and ...
4
votes
0answers
73 views

“Edge Density” of Infinite Planar Graphs

Given an infinite planar graph $G$, let's denote by $\{H_1,H_2,\dots,H_m\}$ all the labeled graphs on $n$ vertices that appear as subgraphs of $G$. Also let $$d_n=\frac{\sum_{i=1}^m \#E(H_i)}{nm}$$ ...
1
vote
0answers
169 views

“Graph Individualization”[ reference request] [on hold]

Context: I am studying Weisfeiler Lehman method(WL method) and have clear idea about 1 and 2 dimensional WL method. I was wondering about the individualization process described below- ...
2
votes
1answer
131 views

inequality with exponents

We are given a graph $G$, each vertex $v$ has an assigned value $\gamma_v\in [0,1]$, and it happens that for every $v$ we have $\gamma_v+\sum_{u\in \delta(v)} \gamma_u = 1$. Assume that $\sum_v ...
2
votes
1answer
110 views

Embedding graphs into hyperbolic spaces

Do we know of a characterization as to when does a graph have a "good" embedding into a hyperbolic space? (And does having such an embedding have a spectral or wavelet analysis signature?) I don't ...
-4
votes
0answers
171 views

Has Frucht's theorem been successfully used in inverse Galois theory? [closed]

Logically, one can associate to any finite extension $K$ of $\mathbb{Q}$ a directed graph describing it. Can such a graph be used together with Frucht's theorem asserting that every finite group is ...
3
votes
1answer
187 views

Can every permutation group be realized as the automorphism group of a graph (acting on a subset of the vertices)?

By Frucht's theorem, every finite group can be realized as the automorphism group of a finite undirected graph. Because a permutation group is a finite group, it is clear that every permutation group ...
3
votes
0answers
86 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 ...
1
vote
0answers
65 views

Efficient realization of a restricted neighborhood hypergraph

In a graph $G$, the (open) neighborhood of a vertex $v$ is the set of vertices adjacent to $v$ (so vertex $v$ is excluded). The neighborhood hypergraph of $G$, denoted by $\mathcal{N}(G)$, is a ...
3
votes
2answers
184 views

Hedetniemi's conjecture for graphs with countable chromatic number

Are there graphs $G, H$ such that $\chi(G) = \chi(H) = \aleph_0$, but $\chi(G\times H) < \aleph_0$?
9
votes
0answers
137 views

What algebraic structures are related to the McGee graph?

Recall that an $(n,g)$-graph is a simple graph where each node has $n$ neighbors and the shortest cycle has length $g$, while an $(n,g)$-cage is $(n,g)$-graph with the minimum number of nodes. The ...
1
vote
1answer
65 views

Categorical product of graphs and chromatic number

Let $(G_i)_{i\in I}$ denote a family of simple, undirected graphs (finite or infinite). Let $\prod_{i\in I}G_i$ denote their categorical product. Why do we have the inequality $$\chi(\prod_{i\in ...
1
vote
0answers
78 views

3-regular (cubic) graph with adjacency eigenvalue 1

Suppose $A\in\{0,1\}^{n\times n}$ is the adjacency matrix of a 3-regular (cubic) graph $G=(V,E)$; that is, all $n$ vertices $v\in V$ in the graph have three neighbors. Is there a nice necessary ...
2
votes
2answers
129 views

Detecting HNN-Extension and free products with amalgamation

This question is partly connected with the following Connection between Stalling's end theorem and Seifert-van Kampen Theorem. By Stalling's Theorem a group with more than one end splits over a ...
3
votes
1answer
73 views

Good broad review of agent-based modeling? [closed]

Trying to find some good review of agent-based models and networks, specifically models that are defined by a graph of interacting nodes, that covers analysis of collective behavior based on model of ...
1
vote
0answers
72 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 ...
1
vote
1answer
199 views

Can we estimate the probability $\mathbf{P}(a-k|a - b) $ on a random graph?

Let $G=(V,E)$ be an undirected random graph such that $V$ is the set of nodes, and $E$ is the set of edges Assume the ground graph $G$ is sparse enough, for example, $\frac{|E|}{|V|}= c \in [10, ...
0
votes
1answer
88 views

Reference request: Strong Connectivity and the Incidence Matrix

Question: What would be a good reference for characterizations of strong connectivity of a digraph in terms of its incidence matrix? Details: Consider a digraph $(V, E)$ with vertex set $$V = ...
9
votes
2answers
306 views

Is every metric space quasi-isometric to a graph?

I've proved that if $(X, d)$ is a geodesic metric space then there exists a graph which is quasi-isometric to $X$...during this proof I've precisely used the fact that given two point in $X$ there ...
5
votes
0answers
69 views

Diameter of the modified bubble-sort graph

The modified bubble-sort graph is the Cayley graph $Cay(S_n,S)$ of $S_n$ generated by $n$ cyclically adjacent transpositions. Thus $S = \{ (1,2),(2,3),\ldots,(n,1)\}$. I was wondering whether the ...
0
votes
1answer
74 views

Forbidden Tripartite Graphs

I was looking at extremal graph theory. I have understood the proofs of upper bounds for the Zarankiewicz problem which basically states: What can you say about the edges of a graph with $n$ vertices ...
1
vote
0answers
14 views

Properties of Optimal k-Tour Spanners

Let the edge set of a Optimal k-Tour Spanner of a graph $G$ be equal to the edges of $G$ that lie on at least one optimal tour through exactly $3<k<n$ distinct vertices of $G$. I would like ...
1
vote
0answers
103 views

Classification of Automorphism set of a Regular graph

Let $A$ be the adjacency matrix of an $r$-regular graph $G$ with $n$ vertices (Not complete or cycle graph) . Also, let $Aut(G)$ be the set of all its automorphisms (i.e. set of permutation ...
10
votes
3answers
451 views

Self-Similar Graphs

Many fractals can be generated using and infinite sequence of graphs. For example, Sierpinski's Gasket could be generated by the following sequence of graphs. Many definitions of fractal dimensions ...
17
votes
0answers
496 views

Reference request: Parallel processor theorem of William Thurston

Sometime in the 1980's or 1990's, Bill Thurston proved a theorem regarding the existence of a universal parallel processing machine, using a certain class for such machines having finite deterministic ...
3
votes
0answers
61 views

Relationship of Weisfeiler-Lehman algorithm to weak isomorphism of coherent algebras

A coherent algebra is a matrix algebra (over $\mathbb{C}$) closed under conjugate transpose and Schur (entrywise) product, and that contains the identity matrix $I$ and all ones matrix $J$. Given ...
0
votes
2answers
60 views

Connectivity of weighted graph and zero Laplacian eigenvalues

Given an undirected graph $G$, and let $V$ denote its set of vertices and $E$ its set of edges. Suppose that there are no edges connecting the same vertex, and no more than one edge connecting any ...
5
votes
1answer
300 views

Terminology in combinatorics

I met the following two combinatorial concepts during a study outside of combinatorics. I am wondering if there are common terminologies in combinatorics. A finite graph $G$ has the following ...
1
vote
0answers
35 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 ...
6
votes
1answer
69 views

Analysis of the Laplacian of a random bipartite graph

My analysis of an engineering problem reduced to analysis of the Laplacian of a (random) bipartite graph. There are a few particular questions I am interested in, but not sure which direction to take ...
4
votes
2answers
193 views

A generously vertex transitive graph which is not Cayley?

A graph is vertex transitive if $x \mapsto y$ by an automorphism. A graph is generously vertex transitive if $x \mapsto y \mapsto x$ by an automorphism. Simple facts: GVT $\rightarrow$ unimodular. ...
2
votes
0answers
234 views

Partitioning graph for Graph Isomorphism [on hold]

Motivation: I am studying the graph isomorphism problem. I am trying to construct a partitioning method to reduce search cases . Construction: $G$ is an $r$ regular graph, $k$ connected (not a ...
0
votes
1answer
128 views

Decomposition of a regular graph and connected subgraphs

I have asked almost same question earlier. I have been told that my question was poorly written, so I am trying to write it more clearly in this post. Also, this time I would be a little different in ...
2
votes
0answers
56 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 $t+1$ ...
5
votes
0answers
96 views

Does squaring a directed random graph more than double its out-degree?

As far as I know, it is an unsolved question whether or not this is true: If $G$ is a directed an oriented graph, $G^2$ always has some node whose outdegree is at least double that of its ...
0
votes
0answers
71 views

A constrained minimum edge coloring

Is minimum number of colors needed to color edges of complete graph $K_n$ so that every even simple cycle contains at least one color assigned to odd number of edges at most $\beta n$ where ...
2
votes
2answers
53 views

Minimal edge color on constraints

Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every even simple cycle contains an odd number ($>1$) of colors much larger than $(\log n)^\beta$ or ...
1
vote
0answers
101 views

Efficiently counting all paths of length n in a graph with vertex visitation contraints [closed]

I have a graph G with two classes of vertices. The first class represents no resource limitation entities and can be visited an unlimited number of times in any path traversal. The second class of ...
2
votes
2answers
134 views

“Nice” and “nasty” partitions in graphs

Let $G=(V,E)$ be a simple, undirected graph, that is $V$ is a set and $E \subseteq [V]^2 = \{\{v,w\}: v,w \in V \land v\neq w\}$. For $v\in V$ and $S\subseteq V$ we set $$N(v,S) = \{w\in S: \{v,w\} ...
1
vote
0answers
32 views

One sided Satisfactory Partition problem

The Satisfaction Partition problem is to decide if a graph has a vertex partition (U,V) into non empty parts where each vertex has as many neighbours in its part as in the other part. This problem is ...
1
vote
0answers
367 views

Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition

Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected. The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...
1
vote
0answers
35 views

Weak law for component count of Erdos-Renyi random graphs

Penrose and Yukich derive a weak law for functionals of binomial point processes, which implies a law of large numbers for the component count of random geometric graphs. Do similar results exist for ...
8
votes
2answers
232 views

Dividing the edges and diagonals of a polygon among disjoint sub-polygons

Let $P$ be a convex $n$-gon ($n$ is odd and $n \geq 5$). Determine the smallest $m$ such that all edges and diagonals of $P$ can be covered by the edges of $m$ convex sub-polygons of $P$ which ...