A simple graph (no loops, no multi-edges) $g$ is 'local', when the neighbourhood graph of each vertex is isomorphic to a specific graph $X$. Specific definitions may differ whether the 'central vertex' is included in that neighbourhood but that does not make a difference to the definition of a graph being local.
More generally/ a bit weaker: A graph is 'locally $X$', when the neighbourhood graph of each vertex is isomorphic to a graph family $X$.
Examples of local graphs are the
- the cycle graphs
- the octahedron
- the icosahedron
- highly regular triangulations, e.g., where the neighbourhood of each vertex excluding that vertex is (isomorphic to) $C_6$
- all triangle-free, regular graphs
- all vertex transitive graphs
- the complete graphs
I wonder whether this property of 'locally the graph looks the same everywhere' can be 'seen' or 'heard' in the spectrum of the combinatorial Laplacian ($L=D-A$) of the graph.
What is known about the spectrum of the combinatorial Laplacian of such a graph?
Intuitively, I would expect some sort of smoothness of the spectrum… but that is not quite the case already for the complete graphs… and the set of local graphs may still be too big a set to make more specific statements. I guess I'm a bit on a fishing expedition:
What local property of a graph has a 'smoothening' impact on the spectrum (small gaps between eigenvalues, no peaks in the multiplicity) of the combinatorial Laplacian of a graph?
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