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Results tagged with graph-theory
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user 1266
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.
2
votes
connected and vertex-transitive prime graphs with respect to Cartesian product
It's all in Hammock, Imrich and Klavzar "Handbook of Product Graphs". The rough summary is that everything works nicely, and you do not need transitivity. The automorphism group will be the direct pro …
- 12.1k
6
votes
Is there any partition of a regular graph which in any part there exists a vertex with all i...
You are asking for a perfect 1-code, there is a largish literature. There is no characterization of the regular graphs which contain a perfect 1-code, but a useful necessary condition is the that the …
- 12.1k
5
votes
Accepted
Request for examples of 4-regular, non-planar, girth at least 5 graphs
A random 4-regular graph will have large girth and will, I expect, not be planar. This suggests that that there are a lot of the graphs you want, and they have no particular special properties. Markus …
- 12.1k
1
vote
What is the number of the ways of travelling through a path graph to reach a node from another?
In this particular case there is a simple approach. Take the infinite path with the integers as its vertex set. The number of walks of length $k$ starting at $0$ is $2^k$; the number of walks of lengt …
- 12.1k
0
votes
Find all the isomorphisms between two planar graphs
If $\psi$ is a fixed isomorphism from a graph $G$ to a graph $H$, then each isomorphism from $G$ to $H$ is the composition of $\psi$ with an element of $\mathrm{Aut}(H)$. So I can specify the set of i …
- 12.1k
7
votes
What is the independence number of hamming graph?
I take the Hamming graph $H(d,q)$ to be the Cartesian product of $d$ copies of the complete graph $K_q$. Its independence number is $q^{d-1}$.
Proof: The Hamming graph lies in the Hamming scheme, so …
- 12.1k
3
votes
Accepted
Factors of Kneser graph
Doerfler and Imrich and (independently) MacKenzie showed that any connected graph has a unique factorization into graphs prime relative to the strong product. It follows that if a connected graph is n …
- 12.1k
1
vote
Spectral Graph Theory
The short answer is that, if your graph is not regular, there is no relation. The effect on the eigenvalues of adding a diagonal matrix is the same as adding an arbitary symmetric matrix.
- 12.1k
3
votes
Shannon capacity determined by $\alpha(G)$ and $\chi^*(\bar{G})$???
I will work with the complements. We first want a graph $G$ such that $\omega(G)$ (the maximum size of a clique) is equal to $\chi_f(G)$, its fractional chromatic number. Take $G$ to be the Kneser gra …
- 12.1k
3
votes
power of adjacency matrix
Call a walk in $X$ reduced if it does not contain any subsequence of the form $uvu$, and let $p_r(A)$ denote the matrix whose $uv$-entry is the number of reduced walks from $u$ to $v$. Let $\Delta$ be …
- 12.1k
5
votes
Ihara zeta and chromatic number of graphs
For regular graphs the Ihara zeta function is determined by the spectrum of the adjacency matrix, and so graphs can have the same zeta function and different chromatic number.
For examples take the co …
- 12.1k
4
votes
Accepted
Graphs where every two vertices have odd number of mutual neighbours
Take a Steiner triple system on $v$ points. Let $X$ be the graph with the $v(v-1)/6$ triples
as its vertices, two triples adjacent if the have exactly one point in common. We need
$v\equiv1,3$ modul …
- 12.1k
3
votes
Accepted
Conditions for subgraph relationship in circulant Cayley digraphs
I very much doubt that there is a nice answer for this.
I suspect that this question
is not essentially easier than the more general problem, where we allow $X$ to be any
tournament. If $n$ is a pri …
- 12.1k
4
votes
Accepted
Lovasz theta function integrality
No. For some examples see When the Lovász theta-function saturates its upper bound
- 12.1k
3
votes
Accepted
Showing two vertices have same degree under a certain condition
I'll take $\lambda=1$ and use $E_j$ for $\mathcal{P}_j$. The $k$-th time derivative of $e^{-it(L-E_j)}s$ at $t=0$ is
\[
(-i(L-E_j))^k s.
\]
Now $(L-E_j)s = -v_j$ (because $Ls=0$) and, noting that $E_j …
- 12.1k