Uniform Shared Neighborhood Structures in Edge-Regular Graphs
Jared DeLeo
TL;DR
This paper investigates the properties of uniform shared neighborhood structures in edge-regular graphs, exploring their existence, characteristics, and implications for graph theory, including forbidden configurations and behavior in graph products.
Contribution
It introduces the concept of uniform shared neighborhood structures in edge-regular graphs and analyzes their properties, including forbidden structures and their behavior in graph products.
Findings
Identification of USNS-forbidden graphs
Characterization of USNS in graph products
Insights into the structure of edge-regular graphs
Abstract
A shared neighborhood structure (SNS) in a graph is a subgraph induced by the intersection of the open neighbor sets of two adjacent vertices. If a SNS is the same for all adjacent vertices in an edge-regular graph, call the SNS a uniform shared neighborhood structure (USNS). USNS-forbidden graphs (graphs which cannot be a USNS of an edge-regular graph) and USNS in graph products of edge-regular graphs are examined.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · Topological and Geometric Data Analysis