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Results tagged with graph-theory
Search options questions only
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user 108884
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.
3
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1
answer
198
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Are there many self-complementary perfect graphs?
A graph is perfect if it has no induced subgraph that is either an odd cycle (other than a triangle) or a complement thereof (note that the class of perfect graphs is closed under graph complement). A …
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11
votes
1
answer
609
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Representations of the automorphism group of graphs via spectral graphs theory
Given a (simple) graph $G=(V,E)$ with $V=\{1,...,n\}$ and let $A$ be its adjacency matrix.
I am interested in the representation theory (over $\Bbb R$) of the automorphism group $\def\Aut{\mathrm{Aut …
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8
votes
2
answers
388
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Can two non-equivalent polytopes of same dimension have the same graph?
By a polytope I mean the convex hull of finitely many points. The graph of a polytope is the graph isomorphic to its 1-skeleton. By equivalence of polytopes I mean combinatorial equivalence, i.e. thei …
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16
votes
3
answers
715
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Can we realize a graph as the skeleton of a polytope that has the same symmetries?
Given a graph $G$, a realization of $G$ as a polytope is a convex polytope $P\subseteq \Bbb R^n$ with $G$ as its 1-skeleton.
A realization $P\subseteq \Bbb R^n$ is said to realize the symmetries of …
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5
votes
0
answers
81
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When does the ΔY-family of a simple graph contain multigraphs?
Given a graph $G$, its ΔY-family is the smallest family of graphs that contains $G$ and is closed under ΔY- and YΔ-transformations.
Since YΔ-transformations can introduce multi-edges, the ΔY-family of …
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5
votes
1
answer
197
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Given a 3-connected graph $G$, is there an edge $e$ so that both $G-e$ and $G/e$ are still 3...
Let $G$ be a 3-connected (simple) graph other than $K_4$. In Diestel's "Graph Theory" Section 3.2 we find
Lemma 3.2.2. There is an edge $e$ so that $G\mathbin{\dot-}e$ is still 3-connected (where $G\ …
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2
votes
0
answers
126
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Are there half-transitive convex polytopes?
I only consider convex polytopes, i.e. convex hulls of finitely many points. The (edge-)graph of a polytope $P\subseteq\Bbb R^d$ is the graph consisting of the polytope's vertices, two are adjacent if …
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1
vote
0
answers
145
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Symmetric subgraph configurations
Let $G,H$ be two simple graphs. Let's call a subgraph of $H$ that is isomorphic to $G$ a $G$-subgraph. Consider the following construction:
Construction: Let $\mathcal G=\mathcal G(G,H)$ be a graph d …
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10
votes
3
answers
497
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Given the skeleton of an inscribed polytope. If I move the vertices so that no edge increase...
Let $P\subset \Bbb R^n$ be an inscribed convex polytope, that is, all its vertices are on a common sphere of radius $r$.
Let $G$ be the edge-graph of $P$. For convenience, assume $V(G)=\{1,\dotsc,s\}$ …
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5
votes
0
answers
146
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How to construct 4-regular graphs with few Hamiltonian decompositions?
A Hamiltonian decomposition of a finite simple graph is a partition of its edge set so that each partition class forms a Hamiltonian cycle. This is only possible if the graph is $2k$-regular.
Interest …
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6
votes
1
answer
251
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An eigenvalue upper bound for 1-walk-regular graphs
Let $G$ be a graph and suppose that $G$ is 1-walk-regular (or, if you prefer, vertex- and edge-transitive, or distance-regular).
Let $\theta_1>\theta_2>\cdots>\theta_m$ be the distinct eigenvalues of …
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3
votes
1
answer
124
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Is a simply connected locally 2-connected complex a union of spheres and planes?
Let $X$ be a (potentially infinite) 2-dimensional simplicial complex. Then each link at a vertex $x\in X$ is a graph.
Question. If $X$ is simply connected and each link is 2-connected (in the sense o …
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3
votes
1
answer
176
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When is a $k$-distance-transitive graph already distance-transitive?
Call a (finite and connected) graph $k$-distance-transitive if its symmetry group acts transitively on the pairs in each one of the sets
$$D_\delta:=\{(i,j)\in V\times V\mid \mathrm d(i,j)=\delta\},\q …
- 13.6k
2
votes
0
answers
60
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Antipodal vertices in spectral graph embeddings
Suppose your are given an antipodal graph $G=(V,E)$, that is, for every vertex $v\in V$ there is a unique maximally distant vertex $v'\in V$.
Under which condistions does the following hold:
If $\the …
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5
votes
1
answer
257
views
A graph similar to the Bruhat graph, what is it called?
The weak Bruhat graph (or 1-skeleton of the permutohedron) $B_n$ can be constructed as follows:
the vertices of $B_n$ are the permutations of the tuple $(1,...,n)$, two are joined by an edge, if they …
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