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Roughly speaking, each vertex represents a 3-jm symbol, the graph is converted to a digraph by assigning signs to the angular momentum quantum numbers j, the vertices are labelled with a handedness representing the order of the three j (of the three edges) in the 3-jm symbol, and the graph represents a sum over the product of all these numbers ...
According to Brooks' theorem every connected cubic graph other than the complete graph K 4 has a vertex coloring with at most three colors. Therefore, every connected cubic graph other than K 4 has an independent set of at least n/3 vertices, where n is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices.
The cube-connected cycles of order n is the Cayley graph of a group that acts on binary words of length n by rotation and flipping bits of the word. [1] The generators used to form this Cayley graph from the group are the group elements that act by rotating the word one position left, rotating it one position right, or flipping its first bit.
It is Hamiltonian, has chromatic number 3, chromatic index 3, radius 6, diameter 6 and girth 7. It is also a 3-vertex-connected and a 3-edge-connected graph. It has book thickness 3 and queue number 2. [1] It can be embedded in the genus-3 orientable surface (which can be represented as the Klein quartic), where it forms the Klein map with 24 ...
A cubic graph (all vertices have degree three) of girth g that is as small as possible is known as a g-cage (or as a (3,g)-cage).The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage. [3]
The smallest cubic graphs with crossing numbers 1–11 are known (sequence A110507 in the OEIS). The smallest 1-crossing cubic graph is the complete bipartite graph K 3,3, with 6 vertices. The smallest 2-crossing cubic graph is the Petersen graph, with 10 vertices. The smallest 3-crossing cubic graph is the Heawood graph, with 14 vertices
It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum. The Wagner graph is triangle-free and has independence number three, providing one half of the proof that the Ramsey number R(3,4) (the least number n such that any n-vertex graph contains either a triangle or a four-vertex independent set ...
Markström's searches found four graphs on 24 vertices in which the only power-of-two cycles have 16 vertices. One of these four graphs is planar; however, the Erdős–Gyárfás conjecture is now known to be true for the special case of 3-connected cubic planar graphs (Heckman & Krakovski 2013)