A simple graph (no loops, no multi-edges) $G$ is 'local', when the neighbourhood graph of each vertex is isomorphic to a specific graph $H$; here, the neighbourhood of a vertex $v$ in a graph $G$ is the subgraph of $G$ induced by all vertices adjacent to $v$, i.e., the graph composed of the vertices adjacent to $v$ and all edges connecting vertices adjacent to $v$.
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.
Alternatively, theses graphs are called 'locally H','neighbourhood regular', 'graphs with isomorphic neighbourhood subgraphs' or 'link graphs'.
I find this an intriguing set of graphs, e.g.,
- it can be easily seen/ constructively proven that the only connected, locally $C_3$, $C_4$ and $C_5$ graphs (using the open neighbourhood definition without the central vertex) are respectively the 1-skeletons of the tetrahedron, octahedron and icosahedron... while for $n=6$, there are infinitely many non isomorphic graphs which are locally $C_n$ (regular triangulations); I don't quite understand $n>6$
- more generally; I would like to better understand, when there are just a finite number of cases or infinite families of these for a given neighbourhood $H$
- is it as simple as the closed neighbourhood $H$ (including the central vertex) containing a spanning star, for a locally $H$ graph $G$ to exist? Is there a counter example?
I'm looking for modern review, ideally publicly available for this type of graph.
References
- https://en.wikipedia.org/wiki/Neighbourhood_(graph_theory)
- https://mathworld.wolfram.com/LocalGraph.html
- https://www.researchgate.net/publication/264968930_Neighborhood_regular_graphs
- https://www.researchgate.net/publication/264959077_Graphs_with_Isomorphic_Neighbor-subgraphs
- For a simple, local graph (induced neighbourhood graphs are isomorphic), what is known or expected about the spectrum (combinatorial Laplacian)?
- Can not find this one publicly: Hell, Pavol (1978), "Graphs with given neighborhoods I", Problèmes combinatoires et théorie des graphes, Colloques internationaux C.N.R.S., vol. 260, pp. 219–223
- Ditto: Sedláček, J. (1983), "On local properties of finite graphs", Graph Theory, Lagów, Lecture Notes in Mathematics, vol. 1018, Springer-Verlag, pp. 242–247, doi:10.1007/BFb0071634, ISBN 978-3-540-12687-4
- 'Which trees are link graphs?' at https://doi.org/10.1016/0095-8956(80)90085-4
- Are there any connected graphs with constant link which are not vertex transitive?
- https://mathoverflow.net/questions/491133/locally-isomorphic-graphs
