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The Petersen graph has adjacency spectrum ${3, 1^{5}, (- 2)^{4}}$

Statement

The Petersen graph has adjacency spectrum

${3, 1^{5}, (- 2)^{4}} .$

Facts & Assumptions

Given: The Petersen graph $P$ on the two-element subsets of a five-element set.

[F1]

Two vertices of $P$ are adjacent exactly when the corresponding two-element subsets are disjoint (The Petersen graph on the two-element subsets of a five-element set, adjacent when disjoint).

[F2]

The adjacency spectrum is the multiset of adjacency eigenvalues (Adjacency spectrum, spectral radius, and cospectral graphs).

Proof

technique · direct

1.1

Fix a vertex $X$ . There are exactly three two-element subsets disjoint from $X$ , so every vertex has degree $3$ . If $X$ and $Y$ are adjacent, then they are disjoint and use four of the five points, so there is no two-element subset disjoint from both; if $X$ and $Y$ are nonadjacent, then they meet in one point and exactly one two-element subset is disjoint from both. Therefore the adjacency matrix $A$ satisfies $A^{2} = 3 I + (J - I - A) = 2 I + J - A$ .

F1algebra

2.1

The all-ones vector $1$ is an eigenvector with eigenvalue $3$ . If $x ⊥ 1$ , then $J x = 0$ , so step 1.1 gives $A^{2} x = (2 I - A) x$ . Thus any eigenvalue $θ$ of $A$ on $1^{⊥}$ satisfies $θ^{2} + θ - 2 = 0$ , so $θ \in {1, - 2}$ . If $m_{1}$ and $m_{- 2}$ are their multiplicities, then $m_{1} + m_{- 2} = 9$ and $tr (A) = 0$ gives $3 + m_{1} - 2 m_{- 2} = 0$ . Solving yields $m_{1} = 5$ and $m_{- 2} = 4$ .

step 1.1algebra

3.1

Hence the eigenvalues are $3$ , $1$ with multiplicity $5$ , and $- 2$ with multiplicity $4$ , which is the stated spectrum by [F2].

step 2.1F2∎

Depends on

Used by

The two-subset model reproduces the Petersen spectrum Example

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Steve Butler, Spectral Graph Theory course notes, lecture 9 (standard reference, not scraped)

The Petersen graph has adjacency spectrum {3,15,(−2)4}{3,15,(−2)4}