# Tagged Questions

**6**

votes

**1**answer

143 views

### A family of skew-symmetric matrices corresponding to cycles in graphs

When investigating loops in Markov chains I ran into the following observation.
A cycle in a graph $G$ with $n$ vertices may be represented by a matrix $\Gamma \in \mathbb R^{n \times n}$ having the ...

**0**

votes

**0**answers

65 views

### Existence of a sequence of (almost) Moore irregular graphs embedded on closed surfaces

Let $S_{g}$ denote the genus $g$ closed orientable surface. I'm interested in disproving the existence of a certain configuration of simple closed curves on $S_{g}$. I'd be happy to go into more ...

**8**

votes

**3**answers

295 views

### Characteristic polynomials of trees and E8

In thinking about constructing manifolds via surgery or plumbing, the following combinatorial problem comes up:
If T is a tree with adjacency matrix A and I is the identity matrix of the same order, ...

**3**

votes

**1**answer

187 views

### Equivalence of Hadamard Graph and Hadamard Matrix

I'm reading Distance Regular Graphs by Brouwer, Cohen, and Neumaier. In section 1.8, they explained Hadamard graphs.
Conversion from a Hadamard Matrix into a Hadamard Graph
An $n$-Hadamard graph $G$ ...

**5**

votes

**2**answers

155 views

### Reflexive (hyperbolic) graphs

Is there an effective description of the graphs such that exactly one eigenvalue (of the conventional adjacency matrix) is $>2$ whereas all others are $\le2$?
By "effective" I mean something ...

**0**

votes

**0**answers

59 views

### Lemma about complete subgraphs of r-parite graphs by Bollobás

Lemma connected with counting z(m,n;s,t): Let $m, \, n,\, s, \, t, \, r, \, k$ be integers, $0 \leqslant s \leqslant m$, $0\leqslant t \leqslant n$,\ $0\leqslant k$, $ 0\leqslant r \leqslant m$ and ...

**3**

votes

**2**answers

301 views

### spectrum of an adjacency matrix

The adjacency matrix of a non-oriented connected graph is symmetric, hence its spectrum is real.
If the graph is bipartite, then the spectrum of its adjacency matrix is symmetric about 0. A few ...

**5**

votes

**2**answers

216 views

### Universal graphs on higher cardinals

The Rado graph contains every finite graph as induced subgraph, and its also holds for countable graphs. So it is an universal graph of size $\aleph_0$, which contains all graphs of size $\aleph_0$ as ...

**1**

vote

**1**answer

175 views

### Counting edges in embeddable CW-complexes in R^3

Using Euler's formula ($V-E+F = 2$ where $V$, $E$ and $F$ are the number of vertices, edges and faces), we can easily count the number of edges in maximal graphs that are embeddable in plane: 3n-6. I ...

**15**

votes

**0**answers

218 views

### Is the Poset of Graphs Automorphism-free?

For $n\geq 5$, let $\mathcal {P}_n$ be the set of all isomorphism classes of graphs with n vertices. Give this set the poset structure given by $G \le H$ if and only if $G$ is a subgraph of $H$.
...

**3**

votes

**0**answers

73 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 ...

**2**

votes

**1**answer

150 views

### Regular graphs with $a$ and $b$ Hamiltonian edges

Special case of this question.
Let $G$ be $r$-regular Hamiltonian graph.
An $a$ edge is an edge which is on every Hamiltonian cycle.
A $b$ edge is an edge which is on no Hamiltonian cycle.
$a(G)$ ...

**5**

votes

**6**answers

408 views

### Random planar, bipartite graphs

I have a need to generate random planar graphs none of which have an odd cycle,
i.e., bipartite graphs.
I know there is a substantial two-decade literature on random planar graphs, little with which I ...

**2**

votes

**1**answer

94 views

### Subdividing toward a unit distance graph in the plane

I just want to ask, what is the significance of subdividing a graph toward a unit distance graph? I have seen several studies of it but I cannot find what its importance.
I mean, like the study of ...

**5**

votes

**1**answer

274 views

### Bipartite Graphs arising from two k-partitions of a given Graph

Let $G$ be an $n$-chromatic connected graph. Let $(V_1, V_2, \cdots, V_n)$ and $(U_1, U_2, \cdots, U_n)$ be two partitions of $V(G)$ corresponding to proper n-colorings of $G$.
Consider the bipartite ...

**7**

votes

**2**answers

329 views

### Graphs with many edges avoided by Hamiltonian cycles

Let $G$ be a $3$-connected Hamiltonian graph with at least one edge that belongs to each H-cycle of $G$. Some authors (e.g. in the link given here) call such an edge an a-edge and an edge that belongs ...

**5**

votes

**2**answers

199 views

### Number of unlabelled planar graphs

What are the best known bounds on the number of non-isomorphic (unlabelled) planar graphs on $n$ vertices? Is there a simple proof that this number is at most exponential in $n$?

**1**

vote

**1**answer

142 views

### Graph classes where Hamiltonian Cycle and Hamiltonian Path problems have different complexity

While searching The information System on Graph Classes and their Inclusions, I stumbled on several graph classes for which the Hamiltonian Cycle problem is $NP$-complete while the complexity of ...

**6**

votes

**0**answers

284 views

### the length of paths in a specific graph

Let $n$ be a positive integer and $K$ be the set of all the $2$-elements subsets of $\{1,2,...,n\}$,then $|K|= \binom{n}{2}$. Define $$S=\{P\subseteq K:\bigcup_{I\in P}I=\{1,2,...,n\}\}.$$
For any ...

**1**

vote

**1**answer

198 views

### Is this Graph parameter known?

Let $\lambda(G)$ denote the edge-connectivity of $G$.
Consider the following parameter:
$\rho(G) = \max_{X \subset V(G)} \min(\lambda(G[X]), \lambda(G[V(G) - X]))$
Has this parameter been studied? ...

**4**

votes

**1**answer

107 views

### Graph Verification Problem

Does anyone know whether the following problem has been solved or has an easy solution?
Given a graph $(V,E)$, two subsets of the vertices $U_1=\{u_1, \dots, u_r \}, U_2=\{v_1, \dots, v_s \} \subset ...

**2**

votes

**0**answers

50 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) ...

**5**

votes

**0**answers

112 views

### Complexity of finding three perfect matchings with no edge in common in a bridgeless cubic graph

According to a conjecture:
Conjecture (Fan & Raspaud, 1994) Every bridgeless cubic graph contains three perfect matchings with no edge in common.
Equivalent statement here
Main question:
...

**13**

votes

**3**answers

639 views

### Are infinite planar graphs still 4-colorable?

Imagine you have a finite number of "sites" $S$ in the positive quadrant
of the integer lattice $\mathbb{Z}^2$,
and from each site $s \in S$, one connects $s$ to every lattice point to which it
has a ...

**2**

votes

**1**answer

149 views

### About an equivalent to Tutte's 5-flow Conjecture

A while back I remember reading that F. Jaeger proved that Tutte's $5$-flow conjecture is equivalent to a statement about the co-planarity of a certain set of points in some euclidean space. But I ...

**3**

votes

**2**answers

277 views

### Which directed graphs have a normal adjacency matrix?

I am working on a problem in matrix analysis and I am looking for certain types of normal matrices. I suspect that these "special" normal matrices arise as adjacency matrices of certain graphs. My ...

**3**

votes

**2**answers

229 views

### Can the Vertices of cubic graph be partitioned into and induced cycle and a forest?

Let $G$ be a $2$-connected $3$-regular graph. Can $V(G)$ be partitioned into $V_1$ and $V_2$ where
$G[V_1]$(the induced subgraph on $V_1$) is a cycle of $G$ and $G[V_2]$ is a forest (Acyclic ...

**2**

votes

**1**answer

139 views

### Is the set of edge of a cubic graph the union of a cycle and and an Acyclic graph?

Let $G$ be a $2$-connected $3$-regular graph. Is it true that $E(G) = E_1 \cup E_2$ where
$G[E_1]$(the induced subgraph on $E_1$) is a cycle of $G$ and $G[E_2]$ is a forest (Acyclic subgraph) of $G$?
...

**2**

votes

**0**answers

81 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). ...

**0**

votes

**1**answer

81 views

### Topological Irreducible graphs for the projective plane

I learned that there are 103 topological irreducible graphs for the projective plane but I am unable to find examples of said graphs. I am unsure of how to find them on my own and I would like to see ...

**4**

votes

**1**answer

120 views

### Counting the number of $(d_v,d_c)$ regular bipartite graphs

I am trying to count the number of $(d_v,d_c)$ regular bipartite graphs. To be specific, let $n,m,d_v,d_c$ be positive integers such that
$$n\times d_v=m\times d_c.$$
Then, what is the number of ...

**2**

votes

**1**answer

181 views

### Coloring of subgraphs of G^n

Let $G=(L,R,E)$ be a finite bipartite graph, such that for each $v\in L\cup R: deg(v)>0$. Define $E^{(n)}=\{(\overline{l},\overline{r}) | \overline{l}=(l_1,...,l_n)\in L^n , ...

**3**

votes

**1**answer

212 views

### Expected Value for a Connected Graph

Consider a connected graph of N nodes.
Assign randomly to each node a distinct number from 1 to N.
For each node consider the maximum adjacent value or itself if all adjacent values are smaller.
...

**1**

vote

**0**answers

131 views

### A connection between nonplanar complete graphs and the alternating group?

I didn't get any response on MSE so I though I'd give this a try here (my question on MSE).
I went to an undergrad's senior honors thesis presentation a while ago. She was discussing crossing numbers ...

**2**

votes

**1**answer

178 views

### Graph of Grassmannian

Let p be an integer, and let G be the graph $(V=Gr(k,\mathbb{F}_q ^n),E)$
where: $Gr(k,\mathbb{F}_q ^n)$ is the set of all subspace of $\mathbb{F}_q$ of dimension k, and $E=\{ W_1,W_2 \in V | ...

**5**

votes

**1**answer

273 views

### A new question about maximal independent sets in regular graphs

This is a question inspired by "A question about independent set in regular graphs".
Suppose that $G$ is a simple $r$-regular graph with $n$ vertices.
We say $H$ is a dominating set for $T$, if
for ...

**11**

votes

**2**answers

380 views

### Strongly connected directed graphs with large directed diameter and small undirected diameter?

This question is an attempt to make progress on domotorp's interesting challenge. This question was originally asked in two parts; the former of which was answered by Ilya Bogdanov, and the latter of ...

**6**

votes

**0**answers

138 views

### Uniformly sampling from the set of all simplicial maps

Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout.
How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial ...

**0**

votes

**1**answer

77 views

### Basis of Cycle Subspace of a Graph

Let $G$ be a $2$-connected graph and for $e \in E(G)$ denote by $\mathcal{C_e}$ the set of all cycles(circuits) of $G$ containing the edge $e$.
For what set of edges does $\mathcal{C_e}$ contain a ...

**-1**

votes

**2**answers

154 views

### spanning tree of a graph of minimum degree three

Does each graph of minimum degree three admit a spanning tree whose vertices have degree three (exactly) except the leaves (degree one)?

**3**

votes

**1**answer

160 views

### Minimum distance between Hamiltonian cycles in cubic Hamiltonian graph

It is $NP$-hard to find constant factor approximation of longest cycle in cubic Hamiltonian graphs. Therefore, finding a Hamiltonian cycle in a cubic Hamiltonian graph is NP-hard.
By Smith's theorem, ...

**0**

votes

**1**answer

247 views

### A question about independent set in regular graphs

Suppose that $G$ is a simple $r$-regular graph with $n$ vertices.
We say $H$ is a dominating set for $T$, if
for every vertex $v\in T$, we have $v\in H$ or there is a vertex $u\in H$ such that $vu\in ...

**4**

votes

**1**answer

246 views

### Fundamental Cycles of a graphs

For a $2$-edge-connected simple graph $G$ and a tree $T$ of $G$, let $C_e$ be the unique cycle in $T + e$, $e \in E(G) - E(T)$. Define the set $\mathcal{C}(T) = \{C_e | e \in E(G) - E(T)\}$.
Now ...

**1**

vote

**0**answers

140 views

### Can assigment of Cayley graphs be functorial?

Let $G$ and $G'$ be finitely generated groups and $f:G\to G'$ a homomorphism. First question: for a given $f:G\to G'$ it possible to select generating sets $S\in G, S'\in G'$ so that their would be a ...

**2**

votes

**1**answer

153 views

### Labeling vertices in a graph

Is there an efficient algorithm for the following task?:
Given a graph $G$, either find a labelling of vertices with bit strings of length $k$ such
that the labels of adjacent vertices differ in ...

**2**

votes

**2**answers

219 views

### Does this graph have a name?

Let $G$ be a connected graph on $n$ vertices and $\mathcal{T}$ be the set of all spanning trees of $G$.
Consider the graph whose vertices are the elements of $\mathcal{T}$
and
$T, T' \in ...

**5**

votes

**1**answer

188 views

### Paley graphs over $p^{2}$ vertices

I have proved that every Paley graph $P(p^{2})$ over $p^{2}$ vertices, where $p\geq 5$ is a prime number has a cospectral mate, i.e. for every prime number $p\geq 5$ there exists a graph $\Gamma_{p}$ ...

**7**

votes

**1**answer

239 views

### Shortest Paths in the “Cantor Graph”

First, let me explain, what I understand by a "Cantor Graph":
it is an infinite, directed graph with self loops and countably many vertices labelled with the natural numbers; every ordered pair of ...

**2**

votes

**0**answers

88 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 ...

**7**

votes

**0**answers

186 views

### Coherence between different ranking methods of a graph's vertices

Given a (connected) graph $G$ it is natural to want to rank its vertices, with the more "central" vertices ranked higher.
Two natural ways of doing it are:
By the degrees.
By the entries in a ...