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, ...

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2
votes
0answers
18 views

Linear intersection number and vertex covering number

A linear hypergraph is a pair $\pi=(V, L)$ where $V\neq \emptyset$ is a set and $L\subseteq {\cal P}(V)$ has the following properties: for $e\in L$ we have $|e|\geq 2$; if $e_1\neq e_2 \in L$ then ...
0
votes
0answers
35 views

Papers about decentralized search and cluster

I just start an independent study about small world network and clusters and I try to find papers about decentralized search and clusters. Can anyone give me some references? Thanks! EDIT (David ...
0
votes
0answers
15 views

About adjacency matrices of $k-$shift lifts of graphs

I am finding the notation of cyclic lifts of graphs to be very confusing. Lets say one is looking at a cyclic $k-$lift of a $\vert V \vert$ sized graph. I would like to understand what is the ...
-4
votes
0answers
66 views

the triangle inequality for shortest paths of graphs [on hold]

In why-the-triangle-inequality I found the statement: for example if $d(a,b)$ measures the "length" of the "shortest path" between points $a$ and $b$ (and this can be interpreted quite ...
15
votes
2answers
326 views

Can all unit-distance graphs have their vertices at algebraic integers?

A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$. Obviously, we can ...
0
votes
2answers
231 views

On independent sets of graph

Given $G$ a regular graph on $n$ vertices, denote $\alpha(G)>1$ to be independence number. Denote $\Gamma(G)$ to be collection of possible subset of independent vertices in $G$ of cardinality ...
2
votes
0answers
81 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 ...
9
votes
2answers
317 views

Can a graph be reconstructed from its cycle lengths?

All graphs discussed are finite and simple. The cycle sequence of a graph $G$, denoted $C(G)$, is the nondecreasing sequence of the lengths of all of the cycles in $G$, where cycles are distinguished ...
5
votes
1answer
226 views

Infinite graphs isomorphic to their line graph

The only finite connected graphs $G$ that are isomorphic to their line graph $L(G)$ are the cycle graphs $C_n$ (see this link for example). There are connected countable graphs that are isomorphic to ...
4
votes
0answers
112 views
+100

Induced subgraphs of small strongly regular graphs

Consider a strongly regular graph $G$ with parameters $(76,30,8,14).$ Hoffman's bound tells us that $\overline{G}$ has an independent set of size at most $4$ and its not hard to see there are indeed ...
0
votes
0answers
36 views

Generating alternating cycles on a perfect matching

Given a perfect matching $M$ in a regular bipartite graph $G$, is there an efficient algorithm to randomly generate self-avoiding alternating cycles with uniform distribution? Ideally, such an ...
0
votes
1answer
84 views

Graph lifts and representation theory

Is there any connection known between the two? One can naturally define lifts of graphs by groups like $\mathbb{Z}_k$ and hence I wonder if representation theoretic properties can be used to say ...
1
vote
1answer
87 views

Graph classes where finding explicit coloring have certificate that it is minumum

Graph coloring doesn't have certificate that smaller coloring doesn't exist in general. I am looking for graph classes where finding explicit coloring is not polynomial and have polynomially ...
-1
votes
1answer
79 views

Maximal independent sets in a graph $G$ versus maximal matchings in the line graph $L(G)$

I'm a bit confused because of the answers in Maximum matchings in infinite graphs . I was thinking that an independent set in a graph $G$ corresponds to a matching in the line graph $L(G)$, and vice ...
2
votes
0answers
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 ...
4
votes
2answers
142 views

Maximum matchings in infinite graphs

For any graph $G=(V,E)$ we define $\mu(G) = \sup\{|M|: M\subseteq E(G) \text{ is a matching}\}$. Is there a graph $G=(V,E)$ such that for every matching $M\subseteq E$ we have $|M|<\mu(G)$?
1
vote
0answers
64 views

Multiple Bipartite graphs and matchings

I've been told recently that it's better i just for help regarding my 'specific' problem rather than lots of little questions around the same topic which appear somewhat unclear. I would first like to ...
2
votes
0answers
59 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]. ...
0
votes
0answers
62 views

When is edge colored circulant isomorphism polynomial?

Don't understand enough group theory, but two papers appear to give partial results about an open problem. Edge colored graph isomorphism is isomorphism which preserves the edge coloring (the ...
0
votes
1answer
81 views

When is a $2$-lift of a graph connected?

Let $\ (V\ E)\ $ be a graph, i.e. $\ E\subseteq\binom V2.\ $ A $2$-lift pattern of a graph is a function $\ e:E\rightarrow\{-1\,\ 1\}.\ $ The induced 2-lift is defined as the graph $\ V\times\{-1\,\ ...
-1
votes
0answers
84 views

Martingales and Bipartite graphs

Would a vertex exposure martingale be useful for bounding the deviation in size of the largest matching from it's expected value in the standard random bipartite graph with vertex classes of size $n$ ...
3
votes
1answer
72 views

Extremal eigenvalues & eigenvectors of skew-adjacency matrix

I am looking for ways to obtain the extremal eigenvalues and eigenvectors of the skew-adjacency matrix of a directed graph. The graphs I am interested in are not regular (but they have a maximum ...
0
votes
0answers
57 views

Signed Laplacians and Ramanujan graphs

Given a signing/2-lift matrix $A_s$ of a $d-$regular graph one has the relationship that the ``Signed Laplacian" is $L = d + A_s$. This $L$ is still the same size as the base graph. But the lifted ...
1
vote
1answer
51 views

Laplacian spectrum of $2-$lifts of graphs

We know that a $2-$ lift of a graph is specified by a $\pm 1$ assignment on the edges of the graph ( given as a signing matrix) denoting which edge is to be duplicated by the identity permutation on ...
2
votes
2answers
130 views

Expected matching in a bipartite graph

Consider a random bipartite graph constructed on vertex classes of size $n$ with each edge present independently with probability $p$. How could I go about calculating the size of the expected ...
5
votes
1answer
135 views

How many cospectral graphs available for a given number of nodes?

Two graphs are said to be cospectral if they have same eigenvalues wrt adjacency matrix, Normalised or Signless laplacian matrix. How many graphs has cospectral mates for a given number of nodes? We ...
10
votes
2answers
372 views

What is the smallest 4-chromatic graph of girth 5?

It is known that the smallest 4-chromatic graph of girth 4 is the Grötzsch graph (11 vertices). What happens for girth 5? The Brinkmann graph (21 vertices) has chromatic number 4, girth 5 and is ...
0
votes
0answers
111 views

Solving gradient of an especial heat equation

In my research I came up with a gradient of heat equation on a edge-weighted graph as: \begin{equation*} \nabla_w T_t(t,w) + T(t,w) . \nabla_w L_w + L_w . \nabla_w T(t,w) = 0 \end{equation*} where ...
3
votes
0answers
102 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
118 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 ...
2
votes
2answers
79 views

Automorphism group of directed complete graph

Given a directed complete graph on $n$ vertices, is there an efficient algorithm for computing its automorphism group? Is there a nontrivial upper bound on the order of its automorphism group? How ...
0
votes
0answers
71 views

What is this graph property: number of vertices it takes to see every vertex?

I am wondering what the name is for the following graph property: given a graph $G$ what is the smallest cardinality of $A\subseteq G$ such that every $v\in G$ is connected to some vertex of $A$? I am ...
0
votes
1answer
124 views

Are these particular kinds of matrices well known?

Given two positive integers $n$ and $a \leq \frac{n}{2}$ consider a $n \times n$ matrix $A$ such that, all the diagonal entries are either $a$ or $a+1$ all the non-zero off-diagonal entries are ...
2
votes
1answer
160 views

Minimum number of edges to remove to have low degree

I have the following problem (k fixed integer): Input: Graph G. Output: Minimum number of edges to remove to G to obtain a graph such that every node has degree at most k. Do you know the complexity ...
0
votes
0answers
97 views

Prove or disprove this upper bound on chromatic number

Let $G$ be a simple connected finite graph and let $v \ge 4$ be the number of vertices, $E$ the number of edges, $\chi(G)$ the chromatic number , $\omega(G)$ the clique number and $\Delta$ the ...
5
votes
0answers
144 views

Sets of spreads in graphs

Let $G$ be a graph. A $k$-spread is a set of cliques of order $k$ which partition the vertex set (so $k|n$, where $n$ is the number of vertices). A partial $k$-resolution of $G$ is a set of pairwise ...
1
vote
0answers
58 views

Hamming graph and independent sets

I'm defining the Hamming graph $H(d,q)$ in the usual way, so we have a set $S$ of $q$ elements, the hamming graph $H(d,q)$ has vertex set $S^{d}$ (the set of all ordered $d$-tuples of elements of $S$) ...
0
votes
0answers
28 views

Maximum flow problem with non-zero lower bound [migrated]

Given G = (V,E ) a directed graph, if $ X \subseteq V $ we note with $\delta ^{+}\left(X\right)$ = $\left \{ xy\in E \mid x \in X, y\in V - X \right \}$ and $\delta ^{-}\left(X\right)$ = $\delta ...
1
vote
0answers
141 views

Threshold for perfect Matchings in Bipartite graph

Consider the random bipartite graph with vertex classes of size $n$ and each edge being present independently with probability $p(n)$. I know one way to prove the threshold of a perfect matching is ...
4
votes
1answer
112 views

Structure of the stabilizer of a vertex-neighborhood of a vertex-transitive graph

Given a simple, undirected graph and a vertex $v$ of the graph, let $L_v$ denote the set of automorphisms of the graph that fixes the vertex $v$ and each of its neighbors. When the graph is ...
0
votes
0answers
59 views

Covering a set in a hypergraph

I'm interested in counting the following. Consider a set $\{v_1,\dots,v_m\}$ of $m$ vertices in the complete $k$-uniform hypergraph on $n$ vertices where $m < k$. I want to know the number of ...
0
votes
0answers
59 views

Parallelism degree of a DAG

Let me first give a motivation. Suppose a connected DAG G with one source X and one sink Y. The goal is to find some "bottleneck" node between X and Y, i.e. node through which every path from X to Y ...
1
vote
1answer
65 views

Spectral radius of a time-varying matrix with strictly positive increment of the matrix's entry

Consider a time varying non-negative matrix $A(t)$ and its spectral radius $\rho(A(t))$ being the largest eigenvalue of $A(t)$ and $t$ denotes the time. If $A(t)$ changes over time with each time a ...
18
votes
5answers
947 views

Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?
10
votes
2answers
563 views

What is a “Ramanujan Graph”?

I noticed an apparent conflict in the definition in literature about what is a "Ramanujan graph, which I was wondering if someone could kindly clarify. (1) The Hoory-Linial-Wigderson review on ...
10
votes
1answer
234 views

Are there non-trivial graphs that uniquely embed to hypercubes?

The $n$-dimensional hypercube is the graph formed by $0$-$1$ sequences of length $n$ where two vertices are adjacent if they differ at only one place. The weight of a sequence is the number of $1$'s ...
2
votes
2answers
191 views

Removing subtrees

Let $T$ be a complete infinite rooted binary tree. Is it possible to remove (infinitely many) subtrees of $T$ and get a subgraph $G$ such that: $G$ has no complete subtrees (the graph below any ...
4
votes
1answer
179 views

Prime labelling of graphs

A prime labeling of a graph is an injective function $f: V(G) \to \{1, 2, ..., |V(G)|\}$ such that for every pair of adjacent vertices $u$ and $v$, $\text{gcd}(f(u), f(v)) = 1$ (labels of any two ...
2
votes
1answer
106 views

How to construct a graph with arbitrarily large girth and large chromatic number? [closed]

Erdos theorem says it is possible and it is not so easy. What is the general procedure to construct graphs like Grötzsch graph?
0
votes
0answers
15 views

Looking to derive bound for modulus of harmonic eigenfunction on weighted graph

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus: $\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$ and wish to lower ...