Hi Chathranee, sure they are!

\begin{theorem}

 For any positive integer $p,q$ the eigenvalues of $K_{p,q}$ are $\sqrt{pq}$, $-\sqrt{pq}$ and 0 with multiplicity $p+q-2$.

\end{theorem}

\begin{proof}

 note that

$$

A(K_{p,q}) = \left(

\begin{array}{cc}

0 & J_{pq} \\

J_{qp} & 0 \\

\end{array}

\right)

$$

where $J_{pq}$ and $J_{qp}$ are matrices of all one's of the appropriate size. Now

$$

rank(A(K_{pq})) = rank(J_{pq}) + rank(J_{qp}) =2

$$

and hence $A(K_{p,q})$ must have precisely two non-zero eigenvalues. These must be of the form $\lambda$ and $- \lambda$, since the trace of $A(K_{p,q})$ is zero. As noted earlier, the sum of the $2 \times 2$ principal minors of $A(K_{p,q})$ is negative the number of edge, that is $-pq$. \\

\indent This sum also equals the sum of the products of the eigenvalues taken two at a time which is $-\lambda^{2}$, thus $\lambda^{2} = pq$ and the eigenvalues must be as asserted in 2. \\

\indent For a positive integer $n \geq 2$. Let $Q_{n}$ be the full-cycles permutation matrix of order $n$. Thus the $(i, i+1)$- element of $Q_{n}$ is 1, $i= 1,2, \cdots, n-1$, the $(n,1)$- element of $Q_{n}$ is 1 and the remaining element of $Q_{n}$ are zero. \\

This completes the proof.

\end{proof}

Thanks Dr. Sunilkumar,

Your answer helped me a lot..

R. Hashimshony, E. Shaviv, A. Wachman, Transforming an adjacency matrix into a planar graph, Building and Environment, Vol 15, No. 4, 1980, p. 205–217

Sometimes it may be a favourable exercise of patience to wait for a reply, before one goes further, and it is another fruitful exercise not to ignore what may bring about further results.

A.S. Asratian, T.M.J. Denley, R. Haggkvist, Bipartite Graphs and their Applications, Cambridge University Press, 1998

The number of perfect matchings in a regular bipartite graph we shall do using doubly stochastic matrices.

Let A=[aij] be an n×n matrix, then the permanent of A, per A, is defined by the formula

                         perA=∑π∈S_{n} a1π(1)a2π(2) …anπ(n)

where S_{n} denotes the set of all permutations of the set {1,2,…,n}.

Proposition 9.3.1 The number of perfect matchings φ(G) of an n by n bipartite graph G is equal to the permanent of its biadjacency matrix B(G). If G is k-regular then φ(G) = kⁿ perA  where  A = (1/k)B(G) is a doubly stochastic matrix.

Theorem 9.3.4 (Egorychev (1980), Falikman (1981)) If A is a doubly stochastic matrix of order n then perA > n!/nⁿ.

From Proposition 9.3.1 and Theorem 9.3.4 we have the following.

Proposition 9.3.5 Let G be a k-regular, n by n bipartite graph. Then the number of perfect matchings of G is at least n!(k/n)ⁿ.

Hi Prof. Wiwat, that's a great answer..I was expecting such a computational property of bipartite graphs... Thanks alot.

Well the spectrum of the adjacency matrix is then symmetric wrt the origin. Not sure if the converse is true.

The relation between bipartite graphs and the eigenvalues of the adjacency are many.

To start:

G is bipartite  iff  

the largest evalue = the negative of the smallest evalue   iff

each evalue lamda sub i = -lamda sub n+1-i       iff

in the characteristic polynomial, every coeff of x^(n-2i-1) must = 0   iff

Every odd moment (the sum of the evalues to an odd power, or if you prefer, the trace of 

the adj matrix A to any odd power) must =0    iff

The trace of any single odd power of A greater than or equal to n is = 0.

There is more, but this theorem, called the pairing theorem, shows the interaction.

Allen Schwenk