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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, topological-graph-theory, random-graphs, graph-colorings and several others.

40 votes
7 answers
9k views

Spectral graph theory: Interpretability of eigenvalues and -vectors

I thought "Wow!" when I learned that the eigenvector of the adjacency matrix of a cycle graph $C_n$ corresponding to the second largest eigenvalue gives the coordinates of the vertices when equally di …
Hans-Peter Stricker's user avatar
34 votes
3 answers
6k views

Bringing Number and Graph Theory Together: A Conjecture on Prime Numbers

Some MOers have been skeptic whether something like natural number graphs can be defined coherently such that every finite graph is isomorphic to such a graph. (See my previous questions [1], [2], [3 …
Hans-Peter Stricker's user avatar
22 votes
4 answers
3k views

Can you determine whether a graph is the 1-skeleton of a polytope?

How do I test whether a given undirected graph is the 1-skeleton of a polytope? How can I tell the dimension of a given 1-skeleton?
Hans-Peter Stricker's user avatar
17 votes
3 answers
996 views

Primacy of arcs/arrows over vertices/objects

Freyd's Abelian Categories is the only textbook I know where the primacy of arrows over objects is taken seriously already in the axioms: there is no talk of objects at all. Only later one sees, that …
Hans-Peter Stricker's user avatar
11 votes
6 answers
1k views

Reasons for the importance of planarity and colorability?

Could it have been foreseen that - exemplarily - planarity and colorability would turn out to be such important concepts in graph theory (there's almost no textbook on graphs without two chapters devo …
10 votes
1 answer
1k views

Reconstruction conjecture: Can other decks do the job?

The standard reconstruction conjecture states that a graph is determined by its deck of vertex-deleted subgraphs. Question: Have other decks been investigated, finding out that only vertex-delet …
Hans-Peter Stricker's user avatar
10 votes
1 answer
897 views

Spanning polytopes

Hamiltonian cycles (seen as spanning polygons) are interesting for several reasons (only a few of which I am aware of), but especially because not every connected graph has a Hamiltonian cycle (is Ham …
Hans-Peter Stricker's user avatar
10 votes
5 answers
883 views

What's the name of graphs with each vertex contained in a cycle?

A tree is a graph with no vertex contained in a cycle. A non-tree is a graph with some vertex contained in a cyle. What's the name of graphs with each vertex contained in a cycle?
Hans-Peter Stricker's user avatar
9 votes
1 answer
911 views

An ubiquitous pattern of questions

There is an ubiquitous pattern of questions concerning assumedly any kind of mathematical object or structure: groups, graphs, numbers, categories, and so on. It goes like this (informally): Can a …
Hans-Peter Stricker's user avatar
9 votes
3 answers
13k views

Number of trees with n nodes and m leaves

Even searching for " 'number of trees' leaves " didn't reveal what I am looking for: an approach for calculating the (approximate) number of trees with exactly n nodes and m leaves. Any hints from MO? …
Hans-Peter Stricker's user avatar
8 votes
7 answers
2k views

Visualizing polyhedra from their 1-skeletons

Except for a few simple cases (typically pyramids and prisms) I find it hard to visualize a polyhedron from its 1-skeleton embedded in the plane, e.g. the hexahedral graph 5, as can be seen here. Too …
Hans-Peter Stricker's user avatar
8 votes
2 answers
2k views

A weaker concept of graph homomorphism

In the category $\mathsf{Graph}$ of simple graphs with graph homomorphisms we'll find the following situation (the big circles indicating objects, labelled by the graphs they enclose, arrows indicatin …
Hans-Peter Stricker's user avatar
8 votes
4 answers
870 views

Self-defining structures

The relations $R$ in abstract graphs (with genuinely propertyless vertices) cannot be defined because there is nothing the relations can base on: they have to be presupposed. But consider derived rela …
Hans-Peter Stricker's user avatar
7 votes
2 answers
1k views

Yet another graph invariant: the similarity matrix

Preliminaries Let $n \in \mathbb{N}$ and $v$ be a vertex of a graph $G$. Let the $n$-neighbourhood of $v$, $N_n(v)$, be the induced subgraph of $G$ containing $v$ and all vertices at most $n$ edges aw …
Hans-Peter Stricker's user avatar
7 votes
1 answer
734 views

Complete tree invariants?

If we take a graph invariant to be "a property that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the graph" (from Wikipedia), I hav …
Hans-Peter Stricker's user avatar

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