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Results tagged with spectral-graph-theory
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Questions related to the spectrum of graphs, defined using one of the possible variants of the discrete Laplace operator or Laplacian matrix. See https://en.wikipedia.org/wiki/Discrete_Laplace_operator
7
votes
0
answers
197
views
Surprising symmetry in the Ramanujan bound
The condition for a connected $(q+1)$-regular graph to be Ramanujan is that every nonzero eigenvalue $\lambda$ of the graph Laplacian satisfy
$$q+1-2\sqrt{q}\le \lambda\le q+1+2\sqrt{q}.$$
With a litt …
- 3,446
4
votes
1
answer
199
views
Digraphs with unique walk of length $k$ between any two vertices
Let $G$ be a digraph such that there is an unique directed walk of length $k$ between any two vertices.
Equivalently, if $A$ is the adjacency matrix of $G$, then $A^k$ is the matrix with all entries $ …
- 3,446
8
votes
0
answers
140
views
Conceptual explanation for the gap in the spectrum of biregular trees
Ramanujan graphs are defined as $(q+1)$-regular graphs for which all nontrivial eigenvalues of the adjacency operator are contained in the interval
$$[-2\sqrt{q}, 2\sqrt{q}].$$
The reason for this def …
- 3,446