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The complete bipartite graph $K_{m, n}$ has adjacency spectrum ${\sqrt{m n}, 0^{m + n - 2}, - \sqrt{m n}}$

Statement

For integers $m, n \geq 1$ , the complete bipartite graph $K_{m, n}$ has adjacency spectrum

${\sqrt{m n}, 0^{m + n - 2}, - \sqrt{m n}} .$

Facts & Assumptions

Given: Integers $m, n \geq 1$ and the complete bipartite graph $K_{m, n}$ with its two parts of sizes $m$ and $n$ .

[F1]

In $K_{m, n}$ every edge joins the two parts, and every such cross pair is an edge (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F2]

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

Proof

technique · direct

1.1

In the vertex order with the first part before the second, the adjacency matrix is $A = (\begin{array}{cc} 0 & J_{m \times n} \\ J_{n \times m} & 0 \end{array})$ by [F1]. If a vector $(x, y)$ has the coordinates of $x$ summing to $0$ , then $J_{n \times m} x = 0$ and $A (x, 0) = 0$ . Likewise, if the coordinates of $y$ sum to $0$ , then $A (0, y) = 0$ . These give an $(m - 1) + (n - 1) = m + n - 2$ dimensional eigenspace for the eigenvalue $0$ .

F1algebra

2.1

On the remaining two-dimensional subspace of vectors constant on each part, $A$ acts by $A (a 1_{m}, b 1_{n}) = (n b 1_{m}, m a 1_{n})$ . Relative to the basis $(1_{m}, 0)$ and $(0, 1_{n})$ , this action has matrix $(\begin{array}{cc} 0 & n \\ m & 0 \end{array})$ , whose eigenvalues are $\pm \sqrt{m n}$ . Together with step 1.1 this accounts for all $m + n$ eigenvalues.

step 1.1algebra

3.1

Therefore the adjacency spectrum is exactly ${\sqrt{m n}, 0^{m + n - 2}, - \sqrt{m n}}$ by [F2].

step 1.1step 2.1F2∎

Depends on

Used by

Two cospectral graphs need not be isomorphic Counterexample
The graph K_3,3 has adjacency spectrum {3,0,0,0,0,-3} Example

Dependency tree · two levels

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Sources

Richard P. Stanley, MIT 18.314 handout, Problem 1 (standard reference, not scraped)

The complete bipartite graph Km,nKm,n​ has adjacency spectrum {mn,0m+n−2,−mn}{mn​,0m+n−2,−mn​}