Knowra Generalized Petersen graph Generalized Petersen graph A generalized Petersen graph is a cubic graph formed from an outer cycle, an inner star polygon, and spokes joining corresponding vertices. It is denoted GP(n,k), where n sets the number of vertex pairs and k sets the inner step.
Cubic graph : A graph in which every vertex has degree three. Every vertex in a generalized Petersen graph has two cycle neighbors and one spoke neighbor.
Vertex-transitive graph : A graph whose automorphisms can map any vertex to any other vertex. Generalized Petersen graphs are vertex-transitive only for particular parameter choices.
Petersen graph : The ten-vertex graph with degree three at every vertex, girth five, and no Hamiltonian cycle. It is the smallest and most famous member, GP(5,2).
Julius Petersen : A Danish mathematician known for foundational work in graph theory. The family takes its name from Petersen's celebrated graph.
Cycle graph : A graph whose vertices form one closed path, with each vertex adjacent to its two neighbors. The outer rim is a cycle, and the inner vertices follow a fixed-step cycle pattern.
Graph automorphism : A permutation of a graph's vertices that preserves adjacency. Automorphisms capture symmetries among the rim, spokes, and inner cycle.
Dodecahedral graph : The graph whose vertices and edges form the skeleton of a regular dodecahedron. It is the generalized Petersen graph GP(10,2).
Graph theory : The mathematical study of graphs, consisting of vertices and edges. Generalized Petersen graphs became a recurring test family for structural questions.
Graph degree : The number of edges incident to a vertex in a graph. The degree-three condition follows directly from the two cycle edges and one spoke at each vertex.
Hamiltonian path : A path that visits every vertex of a graph exactly once. Whether these graphs contain spanning paths or cycles varies with their parameters.
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