The Petersen graph 𝒫 is the graph whose vertices are the ten 2 -subsets ul {1,2,3,4,5} in which

step 1 we can prove this result by induction n. So when n is equal to two vertices, if this graph is co

The Petersen graph 𝒫 is the graph whose vertices are the ten...

The Petersen graph $\mathcal{P}$ is the graph whose vertices are the ten 2 -subsets ul $\{1,2,3,4,5\}$ in which two vertices are joined by an edge if and only if thein 2-subsetss are disjoint. (a) Draw a picture of the Petersen graph. (It can be drawn as a pentagon with a disjoint pentagram inside it-so 10 vertices and 10 edges - where there are an additional five edges joining each vertex of the pentagon to the corresponding vertex of the pentagram.) (b) Verify that for each pair of vertices of $\mathcal{P}$ that are not joined by an edge, there is exactly one vertex joined by an edge to both. (c) Verify that the smallest length of a cycle of $\mathcal{P}$ is 5 .

The Petersen graph $\mathcal{P}$ is the graph whose vertices are the ten 2 -subsets ul $\{1,2,3,4,5\}$ in which two vertices are joined by an edge if and only if thein 2-subsetss are disjoint.
(a) Draw a picture of the Petersen graph. (It can be drawn as a pentagon with a disjoint pentagram inside it-so 10 vertices and 10 edges - where there are an additional five edges joining each vertex of the pentagon to the corresponding vertex of the pentagram.)
(b) Verify that for each pair of vertices of $\mathcal{P}$ that are not joined by an edge, there is exactly one vertex joined by an edge to both.
(c) Verify that the smallest length of a cycle of $\mathcal{P}$ is 5 .

Key Concepts

Cycles and Girth

A cycle in a graph is a closed path where the starting and ending vertex is the same, and none of the other vertices are repeated. The girth of a graph is defined as the length of its shortest cycle. In the context of the Petersen graph, identifying that the shortest cycle has length 5 provides valuable insight into its structure, impacting properties such as connectivity and the graph's resistance to certain types of decompositions.

Petersen Graph

The Petersen Graph is a classic example in graph theory that is known for its unique structural properties and high degree of symmetry. It is often defined through various constructions, including one where its vertices represent the 2-element subsets of a 5-element set and edges connect vertices whose corresponding subsets are disjoint. This graph serves as a counterexample in many scenarios and illustrates many non-intuitive results in graph theory.

Graph Drawing Representations

Graph drawing is the practice of representing the vertices and edges of a graph in a visual format, which can help in understanding its structure and properties. In the case of the Petersen graph, a common depiction is as a pentagon with an inscribed pentagram, a drawing that emphasizes its symmetry and makes features like cycles and connectivity more apparent.

Strongly Regular Graphs

A strongly regular graph has the property that each pair of non-adjacent vertices shares the same fixed number of common neighbors. The Petersen Graph exhibits such a property wherein every two vertices that are not directly connected have exactly one common neighbor. This regularity in neighbor relations not only contributes to the graph's symmetric nature but also places it within a broader class of graphs that are important in spectral graph theory and combinatorics.

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