The Petersen graph is strongly regular (with signature srg(10,3,0,1)). It is also symmetric, meaning that it is edge transitive and vertex transitive. More strongly, it is 3-arc-transitive: every directed three-edge path in the Petersen graph can be transformed into every other such path by a symmetry of the … See more In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The … See more The Petersen graph is nonplanar. Any nonplanar graph has as minors either the complete graph $${\displaystyle K_{5}}$$, or the See more The Petersen graph has chromatic number 3, meaning that its vertices can be colored with three colors — but not with two — such that no edge connects vertices of the same color. It has a list coloring with 3 colors, by Brooks' theorem for list colorings. See more The Petersen graph is the complement of the line graph of $${\displaystyle K_{5}}$$. It is also the Kneser graph $${\displaystyle KG_{5,2}}$$; this means that it has one vertex for each 2-element subset of a 5-element set, and two vertices are connected by an … See more The Petersen graph has a Hamiltonian path but no Hamiltonian cycle. It is the smallest bridgeless cubic graph with no Hamiltonian cycle. It is See more The Petersen graph: • is 3-connected and hence 3-edge-connected and bridgeless. See the glossary See more • Exoo, Geoffrey; Harary, Frank; Kabell, Jerald (1981), "The crossing numbers of some generalized Petersen graphs", Mathematica Scandinavica, 48: 184–188, doi:10.7146/math.scand.a-11910. • Lovász, László (1993), Combinatorial Problems and Exercises (2nd … See more WebThe girth is the length of the shortest cycle. This is probably the easier part of the question. What possible lengths can cycles have? If you just work your way up from the smallest possible length, you should be able to see which actually occur.
Graph Theory 06: Petersen Graph and Radius - YouTube
WebThe Hoffman-Singleton theorem states that any Moore graph with girth 5 must have degree 2, 3, 7 or 57. The first three respectively are the pentagon, the Petersen graph, and the Hoffman-Singleton graph. The existence of a Moore graph with girth 5 and degree 57 is still open. A Moore graph is a graph with diameter \(d\) and girth \(2d + 1 ... WebIn graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star … farm tuff cart
AMS303 GRAPH THEORY HOMEWORK
WebAmong the six incarnations of the Petersen graph, the middle one in the bottom row exhibits just 2 crossings, fewer than any other in the collection. In fact, 2 is the crossing number of the Petersen graph. ... Necessary for the proof is the notion of girth. The girth of a graph is the length of the shortest cycle the graph contains. WebI describe the Petersen graph, define the radius of a graph, and prove that the diameter of a graph can be bounded by twice its radius.The material follows D... WebMar 6, 2024 · Coxeter's notation for the same graph would be {n} + {n/k}, a combination of the Schläfli symbols for the regular n-gon and star polygon from which the graph is formed. The Petersen graph itself is G(5, 2) or {5} + {5/2}. Any generalized Petersen graph can also be constructed from a voltage graph with two vertices, two self-loops, and one ... free software for vinyl design